DOCUMZNT RESUME ED 025 437 By-Zimmerman, Joseph Central Iowa Low Achiever MathematicsProject ESP. Central Iowa Low-Achiever MathematicsProject, Des Moines. Elementary and Secondary Education. Spons Agency-Office of Education(DHEW), Washington, D.C. Bureau of Pub Date (Nov 68] Grant- OEG- 3965 Note- 64p EDRS Price MF-$0.50 HC-$3.30 *Instructional Materials, Descriptors-Curriculum, Curriculum Development,*Elementary School Mathematics, *Low Achievers, *Mathematics,*Problem Solving. SecondarySchool Mathematics Identifiers-Central Iowa Low AchieverMathematics Project The materials in this EnrichmentStudent Project (ESP) aredesigned especially for the low achiever student inmathematics. The booklet is aselfontained unit consisting of fourelements--a mathematicalpuzzle, a set of instructions,response sheets, and a suitable containerfor keeping the unittogether. ESP is amotivational the student's interest andpromoting hisinvolvement in a idea aimed at attracting been collected portion of mathematicsthat can be enjoyed. Thematerials which have for this ESP, complete with solutionof problems for the teacher'sconvenience involve prepared under ESEATitle peg, dissection,cube, and topologypuzzles. This work was III contract. (RP) Central Iowa LowAchiever Mathematics Project
U.S. DEPARTMENT Of HEALTH, EDUCATION & WELFARE
OFFICE Of EDUCATION
THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE ED025437 PERSON OR ORGANIZATION ORIGINATING II.POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION
POSITION OR POLICY.
The work presented or rennrted herein was performed our, suant to a Grant, OE No. 3969, from the U.S. Office of Educe. tion, Deprirtment of Hei Ith. Ed,initi-n, and Welfare.However, the opinions exnrer,sed hrTiin do not necesssrily reflect the positionorpolicy of the U.S. Office of Education, and no official endorsement by the U.S. Office of Education shoultl.be Inferred.
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TABLE OF CONTENTS Page Section
I. INTRODUCTION
PEG PUZZLES 1 1)Ten Men In A Boat 5 2)Reversal 7 3)Quadrix 8 4)Elimination Plus 10 5)Elimination Square 11 6)Elimination Triangle 12 7)Garage Shuffle
III. DISSECTION PUZZTJES 1 1)Tanagrasm 4 2)Dissected Square 6 3)Dissected "T" 7 4)Cross Dissections 8 5)Star Trick 9 6)Butterfly Puzzle 11 7)Hexed 12 8)Voodoo
IV. CUBE PUZZLES 0 1 1) Soma Cube 4 2) Combo 5 3) Gambler's Die 6 4) Paper Cube 7 5) Oriental Cube
V. TOPOLOGY PUZZLES 1 1) Wire Heart 2 2) Shoestring Puzzle 3 3) Kite Knot 4 4) The Gordian Knot 5 5)Another Knot 6 6)Ring Release
VI. MISCELLANEOUS PUZZLES 1 1) Tower of Hanoi 4 2) Puzzling Pyramid 5 3) Double Cross 6 4) Three X Three 7 5) Five X Five 8 6)Stick Puzzles 11 7) Loco 12 8) Digit 13 9) Mystery Maze
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What is it?
An EnrichmentStudent Project is a self-containedunit composed of four sheets, and elements: a mathematical puzzle, aset of instructions, response a suitable container. The puzzle can be almostanything that is a problem commercially made solving situation. There are hundreds of possibilities, some and others hoemade. A few of the possibilities aregiven in the pages which The user follow, buthese are intended only as anintroduction or beginning. 'The instruc- should feelfree to add new ones as well as toalter the old ones. From tions, typd on index cards, should be asclear and as brief aspossible. puzzle, the the instctions a student should be able tolearn the object of the questions that should rules that govern or restrict asolution, and any pertinent
be answered in the process. student The response sheetis just a blank half-sheetof paper which the questions. And, uses,if he so desires, to record hissolution or to answer the enclosed in a finally, the puzzlefinstructioncardsfand response sheets are manilla envelope, orother small box suitable container. A cigar box, shoe box, element of mystery tothe makes it possible to keep theunit together and adds an
puzzle.
Why use it?
Student As the name wouldimply, one of the reasonsfor using Enrichment But more than that Projects is to enrich the student'sstudy of mathematics. interest and involv- E S P is a motivational idea aimed atcatching the student's One of the biggest ing him in a portion ofmathematics that can beenjoyed. attitude many studens problems any mathematics teacherfaces is the negative dealt with before have towards mathematics ingeneral. This attitude must be interest has been arousedby any learning cantake place. Once the student's 2 the puzzle, then the related mathematics can be drawn out.Also, E S P serves as an incentive for students to work,since it can be used as a reward when the normal work has been completed. And, finally, E S P is a way of providing for the individual differences of the students. There are many puzzles where a slow student is on an equal footing with a good student when it comes tofinding a solution, and as a result is in as good aposition to find success. In summary, then, E S P is a way of enriching, motivating, involving, changing attitudes, rewarding, and providing for success.
How can it be used?
There are as many different ways of using E S P as there are teacherswho have used it. The following are but a few of the ways which teachers havefelt were successful.
First of all, E S P can be used on an individual basis, when a student finishes an assignment before the majority of the class and asks,"What can I do now?" Keeping the set of E S P's in a convenien corner will make it possible for a student to get the teacher's permission and to find apuzzle with as little confusion as possible. Requiring a student to get permission first discourages a student from rushingthrough anddoing his work in a slipshod manner.
Some teachers keep a chart with a class list on one sideand the list of
E S P's across the top. Each student is assigned one of the puzzlesand when he has solved it a check mark is recorded.The chart serves as an incentive to master a given puzzle and in so doing to receiverecognition. The same kinl of
recognition can be given by putting up signs on abulletin board naming the first
one to solve a given puzzle orthe champion on one of the puzzles, forexample:
Mary Kay -- 1st to solve the Tower ofHanoi with 8 discs, or John Brown --
20 Men in a Boat.
Secondly, E S P can be used with an entire class. Sometimes the whole
class needs a pick-me-up or a shot-in-the-armbefore anything can be accomplished. If you have a classroom set ofEnrichment Student Projects you can have them passed out and allow fifteen or twenty minutes for each studentto become familiar with a different puzzle. I is xecreational and the students enjoy it, but it is also mathematical and the students learn from it.
Some of the E S P's are such that the same puzzle can beused by the whole class at the same time. Most of these involve the student in makingthe puzzle first, and then in trying to solve it. This method emphasizes the importance of following instructions accurately, and introduces theelement of group competition. Making classroom sets of some of the puzzles such asTower of Hanoi or 10 Men in a Boat could serve as agood shop project for a student whosetalents lie in that direction.
Some teachers leave a few of the games around the room ondisplay so that students will see them and become interested. This allows students to attempt
some of the puzzles before or afterclass and sometimes even before or after
school. The attraction of a puzzle is frequently so greatthat a student will
ask to check out the puzzle so that he cancontinue his search for a solution at
home.
Although each one of the E S P's is completelyself-explanatory, occasionally
it will be advantageous to have one of themexplained to the whole class. This is
also a good way of orienting the class to the useand care of the puzzles. Each
student should understand when and how to usethese ideas. Such a presentation
might be made more effective by having a studentstudy a particularE S P and
then demonstrate it to the class.
E S P's can be used to introduce a unit as mightbe done using the wire
heart to lead into a study of topology. Or, it can be used to showthe applica-
tion of new learnings while in the midstof a unit, as could be done byapplying
base two in the solution of the Tower ofHanoi. And, it can even be used to
conclude a unit as is the case with thedissected square which is to be used at 4 the end of a unit on Euclidean constructions. E S P can even be used as a make-up assignment for a student who has been absent.
Once the idea of E S P has been established in aclass, encourage the students to keep on the look-out for other puzzles which might fitinto the set.
In using E S P in the classroom there are a numberof cautions which should be observed:
1) Don't give out solutions to any of the puzzles.
A student will lose interest if he is told thesolution.
2) Don't be afraid to include puzzles that you are notinterested in or
that you are unable to solve. Students' interests and abilities differ
considerably.
3) Don't over-do it. It is possible to ruin any idea byusing it too long
or too often.
In the pages which follow, a number ofEnrichment Student Projects have
been collected, complete with solutions for theteacher's convenience. An effort
has been made to be as complete as possibletelling how to make it or where to
buy it, giving the instructions which could be cut outand taped on an index
card, and suggesting possible extensions to a wholeclass. Each section is
numbered independently so that additions can be madeconveniently. II: PEG PUZZLES
1) TEN MEN IN A BOAT
2) REVERSAL
3) QUADRIX
4) ELIMINATION PLUS
5) ELIMINATION SQUARE
6) ELIMINATION TRIANGLE
7) GARAGE SHUFFLE
wood playing boards or All the puzzles in thissection can be made on Their object is todiscover a on peg board,using golf tees forplayers. pattern which willlead to a solution. TEN MEN IN A BOAT 1)
each of Materials: A piece of wood 1" x 2" x 12",and ten golf tees, five of two different colors. accommodate Preparation: Drill eleven evenly spaced holesin the wooden "boat" to the ten "men," five of onecolor at each end of the boatleaving the center "seat" empty.A light coat ofsealer completes the project. Enclose the puzzle in a boxwith the cards below.
ESP TEN MEN IN A BOAT P - 1 A few Object: To reverse the positionof the players in as moves as possible.
Rules: 1) You may either jump aplayer as in checkers or move to an adjacentempty seat.
2) You may not jump a playerof the same color.
Can you do it? How many moves does ittake?
.1111110.1.111111.1111111EN, ESP Ten Men In A Boat P 1
Add one more rule to thosegiven on (A):
3) Players may not movebackwards --each player may only movetowards the opposite endfrom which they started.
Does this cut down thenumber of moves in yoursolution?
How many moves do youthink are necessary now? 2
P - 1 ESP Ten Men In A Boat
Put just two players onthe board, one on eachside of the center hole. How many moves arerequired to reverse their positions?
Try the same thing withfour, six, and eightplayers. How many moves does ittake for each?
Do you see a pattern?
Could you predict the numberof moves it would takefor twIlve players?
Can you suggest a rule orformula that will tell the number of moves for any numberof players?
Class Procedure:
1) Introduce the puzzlewith a story like thefollowing: schools, one for In a small midwesterncity there were two exclusive boys decided to arrange boys and one for girls. One day five of the blind dates with five girls atthe other school. condition; that The girls' school consented tothe arrangement on one their chaperon, Miss Lina, must accompanythe group. The girls, knowing sick), suggested Hiss Lina's fear of water(she always got dreadfully sea lake only one boat wasavailable, a boat ride. When they arrived at the Since Miss Lina was a racing canoeseating eleven people singlefile. unwilling to risk getting sick,she contented herselfwith supervising end of the boat and the seating arrangement,putting the girls at one was well out onthe lake the boys at the other. As quickly as the group they decided to exchange seatsto arrive at a moreagreeable arrangement. decided on severalrules: To make more of a challengeof the project, they of upsetting the only one person can change seatsat a time (for fear to pass a fellow boat); it is permissible to moveto an empty seat or pass another canoer, but no girl may passanother girl and no boy may boy. move to theopposite The object of the gameis for all the canoers to to do this in asfew moves end of the boat fromwhich they started and Why would theydecide on such as possible. How many moveswill it take? point? a project? How are the canoersarranged at the half-way sufficient time toexperiment 2) Pass out the puzzles andallow the class If a set of wood with the idea and toformulate a solution. (Note: other such markerscould boats cannot be made,coins or paper clips or arranged in a row.) be used on a paperboat having eleven squares 3) you may 3) It the minimum number of movesis not discovered, add one rule: him never move backwards. If someone does come upwith a solution, ask to demonstrate to theclass how it is done.
4) To analyze the puzzle, putthe table below on theboard or overhead projector and have the classfill in the information byexperimenting. The correct answers are inparentheses.
# of Men # of Moves 0 (0) 2 (3) 4 (8) 6 (15) 8 (24) 10 (35) 12 (48) 100 (2,600) (1/4N2N)
Questions for class discussion: each side of the If there were just twopeople in the boat, one on center seat, how many moveswould it take for them tochange seats? with the How many for four? six? eight? (Make sure the class agrees facts before you put themin the table.)
Is there a patternhere? twelve men? A Can you predict how many movesit would take to move hundred men?
Can you suggest a rulethat will predict thenumber of moves for any number of men? form Y =Ax2+ Bx 5) This puzzle does generate aquadratic equation of the Substitution of the first satof facts shows that C =0. Substitu- + C. Solving tion of the next two setsgives us two equationsin two unknowns. for A and B simultaneouslygives us the equation:
, 2 Y = % X + X
the number of moves where X equals the numberof men and Y equals required to reverse them.
counting the number of movesin each 6) Another pattern canbe observed by you move oneto direction within a solution.For example with two men, the other to theleft passing the right into theempty hole, then you move to the empty seat, atotal of the first one, andfinally one to the right developes. three moves for onepair of players. Notice how the pattern Total Number of Men Men moved in eachdirection = 3 1 pair 1 1 1 = 8 2 pairs 1 2 2 2 1 = 15 3 pairs 1 2 3 3 3 2 1 = 24 4 pairs 1 2 3 4 4 4 3 2 1 1 2 3 4 5 5 5 4 3 2 1 = 35 5 pairs RLRLRLRLRLR 4) The The steps of thesolution for ten menis listed below. Solution: position of the peg you areto move first numberstands for the position to which you areto move and the secondnumber for the the peg in thefifth hole to thesixth it. (5 to 6, means move hole.) Read down, fromleft to right.
A 7 to 5 5 to 6 10 to 9 9 to 7 9 9 to 7 7 to 5 8 to 10 11 to 8 to 9 8 to 7 6 to 8 10 to 11 6 to 8 6 to 8 4 to 6 8 to 10 4 to 6 4 to 6 2 to 4 6 to 8 5 to 4 3 to 4 1 to 2 4 to 6 7 to 5 5 to 3 3 to 1 2 to 4 6 to 7 7 to 5 5 to 3 3 to 2 9 to 7 7 to 5 5 to 3
11111111111111111111s.ii.aim... REVERSAL
Materials: A piece of wood 1" X 6" X 6" and sixteen golf tees, eight each of two different colors. (1) 2 3 Drill seventeen holes in the Preparation: CO piece of wood as in the pattern 4 5 6 opposite. Insert the tees, one color at the top and the other 0 ® 10 11 at the bottom, leaving the 7 8 9 center hole vacant. 0 ® 12 13 11 Enclose the puzzle in a box with the ® ® cards below. 15 16 17
ESP REVERSAL P - A Object: To reverse the position of the players sothat all the bottom ones are moved tothe top and vice versa.
Rules: 1) You may always move to an empty hole orjump a player of a differentcolor.
2) You may never jump a player of the samecolor.
Can you do it?How many moves does it take?
ESP REVERSAL
Add one rule to those on (A):
3) The bottom players may move only to theleft and up, while the top players may moveonly to the right and down.Neither set of players may move backwards.
Does this help you to reduce thenumber of moves you thought were necessary?
Source: Stancraft Products Game: FORE and AFT 1810 Como Avenue Material: Walnut or Oak St. Paul, Minnesota55108 Price: $5.00 6)
to be moved LAution: The first numberindicates the position of the peg REVERSAL may and the second to theposition it is to be moved. be done in forty-six(46) moves. Read down, left to right.
A
3 to 2 10 to 9 9 to 8 17 to 11 9 to 3 8 to 10 11 to 9 16 to 17 15 to 9 7 to 8 14 to 11 10 to 16 12 to 15 9 to 7 12 to 14 13 to 10 6 to 12 12 to 9 6 to 12 12 to 13 9 to 6 6 to 12 5 to 6 6 to 12 6 11 to 9 3 to 6 8 to 5 4 to 4 10 to 11 9 to 3 2 to 8 7 to 9 to 7 8 to 10 15 to 9 1 to 2 9 to 8 16 to15 7 to 1 10 to 9 8 to 10 10 to16 9 to 7 8 to 10 11 to 9 2 to 8 four each of Materials: A piece of wood 1" x2" x 12" and eight golf tees, two different colors. Insert the Preparation: Drill ten holes in the wood asin the pattern above. two sets of tees in analternating pattern leavingthe right hand pair of holes vacant.
4.11MILAIiMftWOMMAIMINOWOMIMMMOOMMOMM1/0.01111=10111MINMINNMOI ESP QUADRIX P- 3
Object: To go from START to FINISHin just four moves.
START FINISH (1060000 0 Rules: 1) A move consists of moving apair of adjacent players, regardless ofcolor, to a pair of adjacent empty holes.
2) You can pass any numberof players moving in either direction
3) You may not reverse theorder of the pair or separate the pair.
and the Solution: The first pair of numbersindicates the pegs to move, second pair, the holes towhich they are to be moved.
2- 3 to 9- 10 5- 6 to2- 3 8- 9 to5- 6 1- 2 to 8- 9 ELIMINATION PLUS 8)
Materials: A piece of peg board seven holes square and four1" square blocks or apiece of wood l' x 8" x 8". Also, 32 golf tees of the same color.
Preparation: Cover or paint the unwanted sections of peg board and glue the blocks on each 0 corner of the bottomto act 14 15 16 17 18 19 20 as legs. Or, if you prefer wood, drill the holes as in the pattern opposite. Insert the tees in all except the center hole.
P - 4 ESP ELIMINATION PLUS
Object: To eliminate all but oneplayer ending with that player in the center hole.
Rules: 1) Beginning with the centerhole empty, jump horizontally and verticallyremoving the jumped players as in checkers.
2) Every move must be ajump.
3) You may not jump diagonally.
How many can you eliminate? Can you get down tojust one? How many moves does ittake?
Sources: Kohner Bros. Inc. Game: Hi Q 155 Wooster Street Material: Plastic New York, New York 10012 Cost: $1.00
Stancraft Products Game: CROSS 1810 Como Avenue Material: Walnut or Oak St. Paul, Minnesota55108 Cost: $5.00 9)
moved and Solution: The first number givesthe position of the peg to be the second number givesthe position to which it is tobe moved, removing the pegjumped each time.
ELIMINATION PLUS can bedone in thirty-one (31 jumps. This is only one of the many possiblesolutions. Read down, beginning with column A.
A
29 to 17 31 to 23 16 to 4 13 to 11 26 to 24 24 to 22 7 to 9 10 to 8 33 to 25 21 to 23 18 to 30 11 to 3 17 to 29 16 to 28 20 to 18 3 to 1 32 to 24 14 to 16 11 to 25 1 to 9 24 to 26 9 to 23 30 to 18 8 to 10 27 to 25 28 to 16 3 to 11 5 to 17 22 to 24 1 to 9 18 to 6 10) ELIMINATION SQUARE
Materials: A piece of wood1" X 6" X 6" and twelve golf tees ofthe same color.
Preparation: Drill thirteen holes inthe wood as in the patternat right. Insert the golf tees leaving the center holevacant.
.0000011010r11/OsetvimprassMell.roesommairowsunmlowomosa...... ESP ELIMINATION SQUARE
Object: To eliminate all but oneof the players ending with the last playerin the center hole.
the Rules: 1) Jump as you wouldin checkers, removing jumped players.
2) You may jump in anydirection provided the holes are adjacentand in a straight line.
Can you do it?How many moves did ittake?
Stancraft Products Game: Bull's-Eye Source: Walnut or Oak 1810 Como Avenue Material: St. Paul, Minnesota55108 Price: $5.00 ELIMINATION TRIANGLE 11)
Material: A piece of wood1" x 6" x 6" and fourteen (14) golftees of the same color.
in the wood Preparation: Drill fifteen holes according to the patternat right. The game looks betterif you cut the wood to make anequilateral triangle. Insert the golf tees leaving the center hole onthe bottom row vacant. Variations are madepossible by leaving other holes vacant atthe start of the game. The most popular variation is toleave a corner holevacant.
P - 6 ESP ELIMINATION TRIANGLE
To eliminate allbut one of theplayers, ending Object: vacant with the last playerin the hole which was at the start ofthe game.
the Rules: 1) Jump as you wouldin checkers, removing jumped players.
2) You may jump in anydirection, provided the holes are adjacent(next to eadh other)and in a straight line. did it How many players can youeliminate?How many moves take you?
Stancraft Products Game: Tri Source: Walnut or Oak 1810 Como Avenue Material: St. Paul, Minnesota55108 Price: $5.00 of the player you areto The first numbergives the position Solution: the hole to move, thesecond number givesthe position of which you are to move.
A 13 to 6 6 to 13 7 to 2 10 to 3 2 to 9 1 to 4 3 to 8 15 to 6 12 to 5 4 to 13 3 to10 14 to12 11 to13 GARAGE SHUFFLE 12)
Material: A piece of wood 1" x 6" x6" or a piece of pegboard and eightgolf P tees, four each of twodifferent colors. 2 3
in r Preparation: Drill the holes in the wood as 5 the pattern opposite. If using pegboard, paint or cover the un- 0 0 6 7 used holes. Insert four golf tees of the same color acrossthe top OL of the "E," the remainingfour across the bottom,leaving the & center four holes empty. 9 10 11 12
ESP GARAGE SHUFFLE P - 7 Object: To switch the cars from oneend of the "E" shaped garage to the other in theleast number of moves possible.
Rules: 1) All paths are one lane, soit is impossible to pass another car.
2) You may move only one car at atime, bnt you may move in anydirection, provided the way is clear.
3) Each car may be moved severaltimes before arriving at its final parking spot. Can you reverse the positionof the cars? How many moves does it take?
the position of the car youare to move, Solution: The first number gives which you are to move. the second numbergives the position to The change is possiblein at least thirty-five moves.
A
7 to 11 7 to 9 2 to 7 8 to 11 4 to 5 5 to 10 5 to 10 10 to 2 7 to 9 1 to 4 2 to 8 2 to 8 9 to 5 5 to 10 5 to l 3 to 7 1 to 7 7 to 9 2 to 7 6 to 3 8 to 3 8 to 1 11 to 7 1 to 8 8 to 7 10 to 2 10 to 2 12 to 8 7 to 1 10 to 2 9 to 5 7 to 10 9 to 12 3 to 6 11 to 5 III: DISSECTION PUZZLES
1) TANAGRAMS
2) DISSECTED SQUARE
3) DISSECTED "V"
4) CROSS DISSECTIONS
5) STAR TRICK
6) BUTTERFLY PUZZLE
7) HEXED
8) VOODOO
and their objectis Dissection puzzles are made outof cardboard or wood of pieces, morethan to reconstruct theoriginal figure. Due to the similarity of the morepopular dissection one combinationis possible. These are only a few of this type. puzzles, but they should serveas anintroduction to puzzles TANAGRAMS 1
Preparation: Draw the patternopposite on a pieceof cardboard 5" x 5" and cut out the pieces with a razor blade or scissors.
Class Procedure: The puzzle is easily constructed and analyzed by a whole class. For instructions, work sheets and a set of silhouettes refer to:
TANAGRAMS@ $.75 from: CILAMP 1164 - 26th Street Des Moines, Iowa 50311
D - 1 ESP TANAGRAMS A These seven geometric pieces,if arranged correctly, can be used to make a varietyof different figures including the ones below. Each figure requires the useof all seven tanagram pieces. Can you make these? Sketch your solutions on a responsesheet.
A 2)
ESP TANAGRAMS D - 1
The four figures below areinteresting because each can be made from any other 1-*v moving just onepiece. Each one uses all seven tanagrampieces. Sketch your solutions on a response sheet.
ESP TANAGRAMS D -
It is also possible to make pairsof congruent (identical) figures using all seven pieces. Are all the ones below possible? Sketch your solutions on a responsesheet.
,11111wwMMIINN, Solutions: 3)
A
I (impossible)
Source: Kohner Brothers Inc. Game: Pythagoras 155 Wooster Street Material: Plastic puzzle and New York, New York 10012 leaflet of silhouettes Price: $1.00
Reference: Tangrams, Ronald C. Read, Dover, 1965. DISSECTED SQUARE 4
Preparation: Draw the pattern atright on a piece ofcardboard and cut out the pieces using a razor blade or scissors.
Project: Have a student make up aset of instructions for making the puzzle using just pencil and ruler, or, perhaps, compass and straight edge (Euclidean constructions). Then, test the instructions by havingthe whole class follow them and when finished, suggested how they might be improved. If you prefer to usesomething which is ready to use, duplicate the instruction sheet(following page). learned the NOTE: These instructions presumethat the student has b) bi- basic Euclidean constructions: a) bisecting a line, The secting an angle, and c)constructing a perpendicular. puzzle is a good way ofconcluding a unit onconstructions.
rIMMOMMONEMIONUMPMIIMMIMMIMOI.1 ESP DISSECTED. SQUARE Every piec These five pieces can beused to make a square. pieces may be must lie flat on yourdesk and none of the Sketch placed on top of anyof the others. Can you do it? your solution on aresponse sheet. just four of the It is also possibleto make a square using pieces. Which piece do youhave to leave out?
N1111111MO.611,111=1~.10111menN.OMMIIM=.111MINPhaVE DISSECTED SQUARE 5)
PUZZLING CONSTRUCTION
Using Euclidean construction procedures, it is possible to make a puzzle composed of five separate pieces. Before you perform the constructions, read through the directions and make a sketch of each piece to decide where each should be placed on a single sheet of paper. After you have finished the constructions, cut the pieces out and see if you can make a square using all five pieces.
Figure A: 1) Draw BC approximately two inches in length. 2) Construct AB 1 BC and AB = BC. 3) Draw AC. Figure B: 1) Draw ST so that it is twice the length of AB. 2) Bisect ST and label the midpoint as X. 3) Construct RS 1 ST and RS = ST. 4) Bisect RS and label the midpoint as Y. 5) Draw RT and XY. 6) Bisect RT labeling the midpoint as Z. 7) Bisect XY labeling the midpoint as W. 8) Draw WZ.
110asowli Figure C: 1) Construct LM = WX. 2) Construct OL 1 LM and NM 1 LM such that both 0 and N are on the same side of LM and that OL =Ia.= MN. 3) Draw ON. Figure D: 1) Construct EF = AC. 2) Construct GE 1 EF and GE = EF. 3) Draw GF.
Figure E: 1) Construct HI = LM. 2) Construct JH I HI and JH = HI. 3) Draw JI. (Use a dotted line--do NOT cut.) 4) Construct FJ 1 JI and FJ = AB. 5) Construct KI 1 IJ and KI = JF. 6) Draw FK.
Cut out triangle ABC, pentagon RSXWZ, quadrilateral LMNO,triangle EFG, and pentagon HIKFJ. See if you can make a square using all five pieces. DISSECTED "T" 6)
Material: A piece of cardboard about 5" x 5"
Preparation: Draw the pattern opposite on a I piece of cardboard and cut out with a razor blade or scissors.
)
-0.11MMIllmolmp.7. OIMMMIIIIIIIPOIMI Nord ESP DISSECTED "T" D - 3
These four pieces can be used to form the letter "T." Can you do it? Sketch your solution on a response sheet. Why was it difficult to make a "T" out of thesepieces? Place an X on your sketch to indicate which piece wasthe keystone or most important in putting the puzzletogether. f..." I \ 1111111>
.IIIIMM CROSS DISSECTIONS 7)
Material: Pieces of cardboard 311 x 511
Preparation: Draw each pattern on a piece of card- board, and cut out with a razor blade orscissors. Each pattern is a separate puzzle.
ESP CPOSS DISSECTION
The pieces to this puzzle can beused to make a regular cross.
Can you do it?
Sketch your solution on a response sheet.
nmememer.1. 8) STAR TRICK
Material: A Diece of cardboard about su x 5H.
Preparation: Draw the pattern on thecardboard and the star. cut out the fivepieces, discarding
VIIIMEMmalat
D - 5 ESP STAR TRICK regular five-point These five pieces canbe used to make star.
What's the trick?
Sketch your answer ona response sheet.
=m11110164114..0.1.111&1=1 BUTTERFLY PUZZLE 9)
Li 11111
(...,) (A) 11
4 4 L...4...i 4
LJ NO (A)
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Material: Pieces of oak tag 6" x 6". oak tag squares,rulers, andscissors. Class Procedure: 1) Pass out the pre-cut into nine equal squares, 2) By measuring,divide the square full piece of three rows of three squareseach using the cardboard. overlay from the 3) Put the pattern onthe board or make an and explain the process pattern and put it onthe overhead, is 3/4" small squares. Each little square of locating the the x 3/4" and each islocated so that2/3 of it is above direction and centered line and 1/3 is belowthe line in one is an excellentproblem in the other. Centering the squares important, since_allure in measurement. This phase is most solution very to center the squareswill make finding a difficult. 10)
4) Number the squares as in the pattern, orreplace each number by an original design or a differentcolor.
5) Cut out the nine squares and see if you canreconstruct the puzzle.
Note: The name BUTTERFLY PUZZLE is due to its originalform; the squares having been butterflies of various colors, each having ahead and a tail. Since butterflies are rather difficult to draw, they werereplaced by squares, circles or triangles. Creative individuals are invLted toreturn the puzzles to its original form or any new formwhich might seem appealing.
ESP BUTTERFLY PUZZLE D - 6
These nine squares can be arranged in a 3 x 3 square,such that each of the figures in the interior is a little square having matching numbers.
More than one solution is possible.
Sketch your solution on a response 41...... 1011.0 sheet. HEXED 11)
Material: Plastic
ESP HEXED
Object: To arrange the twelve pieces(each composed of five square units) into a rectangleeither 6" x 10" or 5" x 12". A three dimensionalversion can be made fromwood blocks, making possible a 3" x 4" x 5" solid.
=nrommonowilma, ...... www..tor AIMIN.0111111110.1.11.1...
puzzlists. Note: The three dimensional versionis much too difficult for most
Source: Kohner Brothers Inc. Game: Hexed 155 Wooster Street Material: Plastic New York, N.Y. 10012 Price: $1.00 VOODOO 12)
Material: Plastic
ESP VOODOO
Object: To use the seven pieces tomake a rectangle and 140 other geometric shapes.
Sources: Kohner Brothers Inc. Game: Voodoo 155 Wooster Street Material: Plastic New York, N.Y. 10012 Price: $1.00
E. S. Low Company, Inc. Game: SHEKO 200 - 5th Avenue Material: Plastic New York, N.Y. 10010 Price: $.49 -"W ...rpowilMOMPPINIMPft.
IV: CUBE PUZZLES
1) SOMA CUBE
2) COMBO
3) GAMBLER'S DIE
4) PAPER CUBE
5) ORIENTAL CUBE
All of these puzzles involve one or morecubes. Each requires for
its solution a sense of form perceptionand an understanding ofhow the
pieces are related. 1
5
Material: Twenty-seven one inch cubes or a set ofchildren's building blocks.
Preparation: Glue the blocks together as in the patternabove.
( 1
[ESP SOMA CUBE C - 1 A These seven pieces can be used to make a cube 3 x 3 x 3. There are supposedly over two hundred solutions to this puzzle.
Can you find just one?
Sketch the position of each piece in your solution on a response sheet. 2)
011MIN/1111111111111110.1111.111111111011111101 ESP SOMA CUBE C - 1
It is possible to make many other figures besides the oube Some are harder than others, but all the ones listed below are possible. See if you cakfigure out how to make one.
amos...rwlmomm.w... ESP SOMA CUBE
Bed Battleship
Pyramid 1 3)
Source: Edmund Scientific Company Game: Soma Cube 300 Edscorp Building Material: Wood Barrington, New Jersey Price: $3.00 08007 and Reference: The 2nd Scientific AmericanBook of Mathematical Puzzles Diversions, Martin Gardner, Simon andSchuster, New York, 1961. COMBO 4)
2 1 2 3 /111111.100,..
4 1
Material: Four plain wood blocks. replace Preparation: Number the faces of the cubes asin the pattern above or each number by a color.
ESP COMBO C - 2
It is posible to arrange these four blocksin a row so that thedigits 1, 2, 3, and 4 appear just once oneach face. The digits may be in any order,for example: 3, 1,2, 4 is a perfectly acceptablearrangement.
Can you do it?
Sketch your solution on a responsesheet, indicating which numbers are on the inside,bottom, back, etc.
1 2 3 4
isomml
Source: Parker Brothers Game: Instant Insanity 10215 Douglas Avenue Material: Plastic Des Moines, Iowa50322 Price: $1.00
4 r .
GAMBLER'S DIE 5)
Material: Nine blocks of wood, 3/4" x 3/4" x 2 1/4" and
I twenty-one notebook ring reinforcements.
Preparation: Paint the blocks red or some other bright color and then stack to make a cube. Glue the circles on the faces and ends of the blocks as in the pattern opposite, or paint white dots instead of the reinforcements.
ESP GAMBLER'S DIE C - 3
These nine blocks can be arranged to make aregular gambler's die (singular of dice). The dots must appear in the correct pattern on each face as shownbelow. To be a correct solution, opposite faces mustalso add up to seven.
Can you do it? orrana.4...... e0111110MWIONNOINI, 000 I o * * 4.111... , PAPER CUBE 6)
Material: Regular notebook paper orwhite plain
paper. R. ow ma/. NOwilo
Preparation: Draw six rectangles,each 2" x 2 3/4". Draw in the 3/8"flaps on each end. Cut out the rectanglesand fold the flaps at a 90 degreeangle as shown below.
ofteso Vv. 4.11.
111M01=K C - 4 'ESP PAPER CUBE build a Using the six pieces of paper,it is possible to strong paper cube. No tape or glue isallowed or necessary. Do the Can you discover howto put the piecestogether? flaps serve a useful purpose? Make sure the flaps arebent at right .... angles to the square andin the same direction. ORIENTAL CUBE 7
This is just one of many such puzzles which are available today. Among the more popular ones are: a barrel, a pig, a mouse, an elephant, a sphere, a truncated cube, etc.
ESP ORIENTAL CUBE C - 5
Object: To discover how to take the cube apart and to remember how to put it back together.
Source: Edmund Scientific Co. Game: Wood Puzzles 300 Edscorp Building Price: $3.50 Barrington, New Jersey 08007 V: TOPOLOGY PUZZLES
1) WIRE HEART
2) SHOESTRING PUZZLE
3) KITE KNOT
4) THE GORDIAN KNOT
6) ANOTHER KNOT
6) RING RELEASE
topological in nature. Each appears The puzzles in thissection are all
where as in factthere is just anoutside. to have aninside and an outside, simplifies the solutionof the others,since The solution of oneof the puzzles These puzzlescould serve as they all work onpretty much the sameprinciple.
of topology such asnetworks, moebiusstrips, anappetizer for other areas
four-color maps, orflexigons. WIRE HEART 1
Material: Heavy wire that will not bend easily.
Preparation: Bend the wire into the figures at right using a pair of diagonals or pliers.
Pay special attention to the size of the loops on the yoke, since the indention of the heart must fit into one of them in order to get the puzzle apart.Also, the loops on the bar must be large enough that theywon't pass through the loopsof the yoke.
T - I ESP WIRE HEART
It is possible to removethe yoke and bar from the wire heartwithout bending, forcing, or taking any joints apart.
Can you do it? Before If so, describe your solution ona response sheet.
After
Tus Sul Source: B & L Gadgets Game: 1421 - 5th Street, N. W. Material: Wire Cedar Rapids, Iowa 52405 Price: $.59
Solution: Insert the indentionof the heart into one of theloops of the yoke as in thediagram and move the bar from oneside of the indentation tothe other. 2 SHOESTRING TROUBLE
3" x 5", a piece Material: A piece of leatherette(upholstery material) about buttons, poker of string or leather shoe laceabout 12" long and two chips, or metal washers.
Cut two slits and a holein the leatherette makingthe hole Preparation: Attach slightly larger than the distancebetween the two slits. the buttons to the ends of theshoe lace and thepuzzle is complete.
T - 2 ESP SHOESTRING TROUBLE
It Is possible to remove thestring from the pieceof leatherette without untieing thebuttons or forcing the buttons through the hole orcutting the string.
Can you do it?
Briefly describe your solution on aresponse sheet.
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the Loie allowing the Solution: Pass the strip formed bythe two cuts through string to be put on or takenoff without removingthe buttons. KITE KNOT 3)
a Material: A piece of string or a leather shoe laceabout 24" long and a piece of wood 1,4"x 4" x 5" or a piece of heavycardboard, and a metal washer or a 1" X 1" X 1/4" piece'of wood.
Preparation: Drill six holes in the piece of wood,which has been cut in the shape of a kite as in the pattern above. Thread the shoe lace through the holes as in the diagram andtie the ends together after passing them through the washer orwood retainer. This piece prevents the knot from beingpulled through the holes in the kite.
ESP KITE KNOT
It is possible to remove the kitestring from the kite without untieing the knot, passing theknot through the holes, or breaking the string.
Can you do it?
Briefly describe your solution on a responsesheet.
Solution: Thread the loop (X in the diagram)through holes A aad B and around the knot, C. Pulling it back to its originalposition will allow the string to be removed completely. THE GORDIAN KNOT 4)
Materials: A piece of wood 1/4"x 3" x 5", or leather, cardboard,floor tile, etc., a piece of string or leather shoelace about 24" long, andfour metal washers or beads that are larger than theholes.
Preparation: Drill three 1/2" holes in the piece of wood.Thread the shoe lace through the holes as in the pattern above,placing two of the washers at the ends of the string and theother two in the middle.
.risararamocalmarm000mpoomb aora...... «....,,..asaysi~~wrewo.awm. +wawa
ESP THE GORDIAN KNOT T - 4
Alexander the Great, as a boy, was given apuzzle to test his cleverness. The puzzle was to untie the GordianKnot. The legend was that whoever could untie theGordian Knot would rule all Asia.
Alexander took one look at the puzzle, took outhis sword and cut the knot.
Can you untie it without cutting it?When the knot is untied, the two washers will come together. You may not untie the ends or break the string.
Describe your solution on a response sheet.
Solution: Thread the loop (X in the diagramabove) through hole A and around the washer. Repeat for hole B. ANOTHER KNOT 5)
Materials: A piece of wood Y. x 2" x 3" or heavy cardboard, a piece of string or leather shoe lace about 12" long, and a metal washer or large button.
Preparation: Drill two 1/2" holes in the piece of wood and thread the string through the holes as in the pattern above, tieing the ends to the washer.
A.11...11110.1110.0.111110*. ESP ANOTHER KNOT T - 5
The object of this puzzle is to remove the string from the holes without untieing the ends of the string, breaking the string, or forcing the washer through the holes.
Can you do it?
Describe your solution on a response sheet.
Solution: Thread the loop (X in the diagram above) through hole B and pass the washer through it. Pulling the loop back to hole A will leave the string free and easy to remove. RING RELEASE 6)
Materials: A piece of wood le x 3" x 4" or a pieceof heavy cardboard, 3metal washers (larger than the holes) and apiece of string or leather shoe lace about 24" long.
Preparation: Drill two le holes in the wood and threadthe shoe string through as in the diagram above,tieing two of the washers onthe ends.
ESP RING RELEASE T - 6
The object of the puzzle is to remove thering (metal washer) from the string without untieing theends, breaking the string, or forcing any of the washers through thJ holes.
Can you do it?
Describe your solution on a response sheet.
Solution: Thread loop X through hole A and passthe washer through the loop. Pulling it back through the holewill release the washer. VI: MISCELLANEOUS PUZZLES
1) TOWER OF HANOI
2) PUZZLING PYRAMID
3) DOUBLE CROSS
4) THREE X THREE
5) FIVE X FIVE
6) STICK PUZZLES
7) LOCO
8) DIGIT
9) MYSTERY MAZE
f -
TOWER OF HANOI 1
sticks 2" in length, Material: A block of wood 1" x4" x 4", three le dowel to small. and six metal washers ofdifferent sizes from large
the block in atriangular pattern. Insert Preparation: Drill three le holes in the dowels and stack upthe washers on oneof the pegs. set made, the Note: If it is not possibleto have a classroom discs on a paper same thing canbe done using cardboard playing board.
TOWER OF HANOI M 1 ESP A to another. Object: Move the pyramidof discs from one peg disc at a time. Rules: 1) You may only move one
2) You may never put alarge disc on top of a smaller one.
Try to do it in asfew moves as possible. discs. It is recommendedthat you start outwith just four As you perfect yoursystem add morediscs.
1111111111111 AINillman 2) TOWER OF HANOI M - 1 ESP the minimum number There is a formulathat will tell you dlscs from one pegto of moves it takestogiove all the It is: LJ another. 2-1 =6 number of discs you are If you fill inthe box with the expression it willgive working with andthen compute the you thenumber of moves,in the triangle.
How many movesdoes it takewith 3 discs? How many movesdoes it takewith 4 discs? with 5 discs? How many movesdoes it take 6 discs? How many movesdoes it take with
Class Procedure: telling the storyof its history: 1) Introducethe puzzle by at a Buddhist Hanoi was discoveredhundreds of years ago The Tower of predicting the endof Hanoi, being used as ameans of monastery near moved a stackof one Around the clockteams of Buddhists the world. needle to anotheraccording to a hundred golden discsfrom one silver when the stackhad beentransferred prescribed ritual. They felt that would end. from one needle toanother, the world explain theobject of the 2) Pass out a setof towers tothe class and puzzle and therules involved:
1. You may only moveone disc at atime. disc on top of asmaller one. 2. You may neverput a large # of to experiment # of 3) After theclass has had time put discs moves and, hopefully,discover a solution, the board andhave the table opposite on (0) The 0 them fill inthe correctnumbers. in the parentheses. correct answers are 1 (1)
pattern? Do you see a 2 (3) would eightdiscs take? How many moves 3 (7) formula 2x - 1 = y,where x is Does the 4 (15) the number ofdiscs and y isthe number the informationin the of moves, fit 5 (31) table? (63) information? 6 I Can Nu use itto find new move a How many moveswould it take to hundred discs? how long would ittake the Buddhists If they moved onedisc a second, to bring aboutthe end of theworld? 3) 4) Some find it difficult toalways move the right disc tothe right especially when the tower thus getting theminimum number of moves, set of number of discs isincreased. To eliminate guess work, a instructions has be, devisedby writing the number ofeach move in l's and base two notation and theninterpreting the position of the O's.
Move the (disc named by the first1, reading from right to left) disc to the of zeros between tower (occuppied by -- if the number the two l's is even) by the (not occuppied by -- ifodd)
(disc named by the second1, reading from right to left) disc. beginning To use this system, we mustnumber the discs 1, 2, 3, .°, with the smallest disc. the 2nd For example, the tenth move,written 1 0 1 0 tells us to move to the tower disc (since the first 1 appearsin the second position) l's is odd) the not occuppied by (the numberof zeros between the two 4th disc (since the second1 appears in thefourth position).
Thus, the first few moveswould be as follows:
1st 1 - move the 1st disc to anempty tower. tower. 2nd 10 - move tne 2nd disc tothe remaining empty the tower occupied bythe 2nd 3rd 11 - move the 1st disc to disc (0 is even)
empty tower. 4th 100 - move the 3rd disc to an the tower not occupiedby the 5th 101 - move the 1st disc to 3rd disc. by the 3rd 6th 110 - move the 2nd disc tothe tower occupied disc.
to the tower occupiedby the 2nd 19th 10011 - move the 1st disc disc.
the tower not occupiedby the 20th 10100 - move the 3rd disc to 5th disc. be written out by Solution: The complete solutionis left up to you, but may continuing the listingabove.
Reference: Kraitchik, MathematicalRecreations. PUZZLING PYRAMID 4
Material: Oak tag or heavy cardboardand scotch tape.
Preparation: Enlarge the pattern below so that the base of the trapezoid is 6" and the sides of the square are 3". Make two pieces, cut out and tape togetherwith the flaps on the inside.
This puzzle can also be made using -cour identical pieces. However, it is the samepuzzle with each piece cut inLalf.
ESP PUZZLING PYRAMID M - 2
It is possible to make a regularEgyptian pyramid (tetrahedran) havingtriangular shaped sides and base.
Can you do it?
Sketch the position of thepieces in your solution on a responsesheet.
..4.101.....101pmwmis.1111100.0...... 11.00...... 1.
Pyramid Puzzle Source: S. S. Adams Co. Game: P. O. Box 369 Material: Plastic (4 pieces) Neptune, New Jersey 07753 Price: $.35 DOUBLE CROSS 5)
Material: Oak tag or cardboard.
Preparation: Cut out eight squares 2" x 2" and number from one to eight. One 54 number on each square.
If you write on both sides of each square, they will 2 8 7 never have to be turned H over.
If you prefer to use a playing board, draw the 53 figure, leaving out the numbers, on another piece of cardboard or on a piece of wood.
4111=0. ESP DOUBLE CROSS M - 3
Arrange the squares as in the pattern below so that no consecutive integers (i.e. 3 and 4) are touching either on a side or on a corner.
The arrangement opposite is not correct because 5, 6, and 7 are touching on a side.
Can you come up with the correct arrangement?
Sketch your solution on a response sheet. rHREE X THREE 6)
Material: Oak tag orcardboard. 2" x 2" Preparation: Cut out nine squares and number themfrom one to nine, one number per square. 8 I Using the same number onboth sides eliminates havingto turn any squares over.
A playing board canbe made by 3 5 7 drawing the pattern of squares opposite on a piece ofcardboard or wood. Thisidea can be ex- tended to larger magic squares ,4 g2 by simply increasingthe number of squares.
The game can alsobe made using painted or pasted on pegboard and golf teeswith the numbers their tops.
1erm.2aba 411.1.111 - 4 ESP THREE X THREE suc Arrange the nine small squaresto make a 3 x 3 square, that the sum of each row,each column, and bothdiagonals is equal.
What is the correct sumfor each row? 2 Is there a patternin your solution that would help you to make a5 x 5, or a 7 x 7 4 5 6 magic square? 9 Sketch your solution on a responsesheet. Is there more than onesolution to this problem?
1 solution to thispuzzle. All The one given aboveis the only unique Solution: both. others are merelyreflections or rotations or
Are-Joy Game Co., Inc. Game: Perfect Squares Source: Plastic and wood 7509 Denison Avenue Material. Cleveland, Ohio44102 Price: $1.00 FIVE X FIVE 7
Material: Oak tag or cardboard.
Preparation: Cut out twenty-five squares 2" x 2" and numberthem so as to end upwith five l's, If you put 5 five 2's, etc. 1 2 3 4 the same number on both sides of a square, itwill never have tobe turned 3 4 over. 4 A playing board canbe made 5 1 out of anotherpiece of 3 cardboard or a piece of wood c marked off with squares as J it is in the patternopposite.
This game can also bemade using pegboard and golf tees with numbers pasted orpainted on their tops.
ESP FIVE X FIVE
Arrange the twenty-fivesmall squares into a5 x 5 square, no4 so that each row,each column, and both diagonals add upto 4 This is a Illau the same number. special type of magic square. 4 What is the sum of eachrow? Ile4 response Sketch your solution on a 4 sheet. Ell Is there a pattern to your solution? 8) STICK PUZZLES the ends cutoff. toothpicks or matchsticks which have had Material: Two dozen small and instructioncards in an envelope or Preparation: Enclose the sticks box and the puzzleis complete.
STICK PUZZLES M 6] ESP A (a coin or paperclip) Arrange foursticks and a marker sticks, turn as in thefigure below. By moving just two the cherry is on the cocktail classupside down so that the outside.
Sketch your solution on a responsesheet.
1
M - 6 ESP E21.(7K PUZZLES represent fences The thirteen sticksin the figure below (stick) were that make sixcattle pens. If one fence or separate removed, could youstill make six pens? compartments?
Sketch your solution on aresponse sheet. 9)
riP0111MIONNIMPIIIII.* STICK PUZZLES M - ESP C equation, can you make a By moving just onestick in each Numerals. true statement? You may use Roman
Record your solution ona responsesheet.
STICK PUZZLES M - ESP D Arrange twelvesticks as in thefigure below. leave just two 1) By removingjust two sticks, squares.
Show your solution on a responsesheet. ID11
three squares 2) Using the samefigure, can you make by moving fourof the sticks?
NIIIMMVamenmplIdpi4,00.1.1mstIMPEOIONIrro,
ESP STICK PUZZLES shown below. Arrange thirteensticks in the pattern others so as tospell out Remove one stickand move three What are matchesmade of? the answer tothe question:
Sketch your answer ona responsesheet. .olomr marasmis ow. mammeampo110141.00 10)
Solutions: A) B)
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1) II = 2) X := II
3) =
D) 1) 2)
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references For other puzzleswhich could be added tothese, check the with an entire class. below. This particular setof puzzles works well Solutions are best demonstrated onan overheadprojector.
Bakst. References: MathematicalPuzzles and Pastimes, Aaron Fun With Puzzles,Joseph Leeming. LOCO 11)
Material: Plastic puzzle...... 1101,1.110111111110...... 1.06...fte omo...... M.1110001011.11,4
wommliamo...... -/000.0.4whimm.m......
ESP LOCO M - 7
Object: To move the large square fromthe upper left hand corner to the lowerright hand corner without lifting any of the other pieces.
rslemmarrorminwwwwwwwftworo.. 4.1111.0. .1
Source: E. S. Lowe Company Inc. Material: Plastic 200 - 5th Avenue Cost: $.49 New York, New York 10010 DIGIT 12)
Material: Plastic puzzle.
(also called: Boss Puzzle and 15 Puzzle)
I..5678.
9 10 11 12
111111111111111110A
11=10MwewarrimommerwmorliW ESP DIGIT M - 8
Object: To arrange the fifteen digitsin various patterns. There are 61 combinations only halfof which are possible.
=110s.
Source: Roalex Company Material: Plastic Forest Hills, New York Cost: $.49 11375 MYSTERY MAZE
Material: Plastic puzzle.
ESP MYSTERY MAZE M 9
Object: To find the quickest way of going fromstart to finish.
Source: Harmonic Reed Corporation Material: Plastic Union Hill Road Cost: $.39 West Conshohocken Pennsylvania 19428