Association of Teachers of Mathematics Chains - Shuffles, Loops and Fixed Points Repeated Instructions and Processes Leading to Interesting Results Chains Shuffles, Loops and Fixed points

Repeated Instructions and Processes Leading to Interesting Results

Bob Vertes

R N Vertes 2020 Guidance These activities are suitable for KS2/3/4 and older. Many were shared at ATM Easter conferences, and some branch meetings, having previously been explored in secondary school classrooms, with student teachers, and INSET with practising teachers. Some are adapted from work shared by David Cain.

Most activities are repeated processes (iteration) with positive integers, usually reaching a fixed point or a loop after a number of steps. Sometimes more than one fixed point, or more than one loop, occurs. The question not to forget to ask is “Why?”

Activities 8, 9, 10, 20, 21, 24, 25 are for KS2 or older Activities 1-7, 13, 17-19, 22, 23 are for KS3 or older, 26-30 more for KS4 Activities 11,12,14-16, are for KS4/5, best with a calculator, Excel or via a computer program

R N Vertes 2020 Guidance While most activities can be explored individually, many lend themselves well to working in pairs or in small groups. A few are ideally explored via People Maths approaches.

Recording results is important; pencil and paper are always needed. Some tasks may require a calculator, and some may benefit from some programming and/or use of Excel. Some thin strips of paper will be needed for the Origami tasks 22 and 23. Large circles are needed for task 20 Circle Patterns and will helpful for illustrating some of the cyclic patterns produced in a number of activities. Squared paper will be needed for some tasks, especially task 21 Spirals, but is also best for recording, including for making own copies of the two suggested tables.

For quite a number of the activities using the “Tens and Ones” table will be helpful including to save time in repetition if results are recorded. Another table is given for Back Front Shuffles, recording the changes in successive lines, adaptable to some other tasks too.

R N Vertes 2020 Repetitive processes - Iteration •ITERATE •ITERATE •ITERATE •ITERATE •ITERATE • ITERATE • ITERATE……

R N Vertes 2020 1. SISYPHUS String • Start with any natural number (positive integer).

• Form a new number counting the number of even digits, the number of odd digits and the total number of digits.

• Repeat the last instruction until you get 2 consecutive numbers the same. What is this number?

• Try again with a different starting number. • Does it matter if you start with an even or odd number? • Why do you think this process is named after Sisyphus?

R N Vertes 2020 2. Divisors and digits also known as Black Hole Numbers

 A. Take any natural number (positive integer) larger than 1.

 B. Write down all the divisors of your number, including one and itself.  C. Find the sum of the individual digits in this list. This is your new number.

 Repeat instructions B and C until you get 2 consecutive answers the same. This the Black Hole Number.

 Try a different starting number, repeat the process. What happens?

R N Vertes 2020 3. Do we always get to one? • (This is the Collatz conjecture, as yet unproved.)

• Start with any positive integer.

• Apply this rule to find the next term in the sequence:- • If the last number is odd, treble it and add 1. • If the last number was even, halve it.

• Continue the sequence until (ideally) you get to 1.

• What patterns can you find?

• Variation: if odd, treble and subtract one; if even, halve.

R N Vertes 2020 4. Variations on the if/then procedure • Start with any positive integer.

• Apply a rule to find the next term in the sequence, and explore the sequences produced:-

• 1. If number is odd, add 3; if even, ÷2. (or: if odd add 5, or 7 ..) • 2. If number divisible by 3, ÷3; otherwise add 1 (or 2, or 4) • 3. If number divisible by 5, ÷5; otherwise add 1 (or 2, or 3, or 4) • 4. If number divisible by 6, ÷6; otherwise add 5 • 5. If number divisible by 7, ÷7; otherwise add 1 (or 2, or 3,4,5,6,)

• Explore other variations...

R N Vertes 2020 5. The 4 digit differences process 1.

• Take a 4 digit number ABCD not all the same digits • The next number in the sequence, also 4 digits, is found by • |A-B| |B-C| |C-D| |D-A| i.e. the magnitudes of successive differences of the digits • The sequence will always at end with 0000, but some numbers take longer than others. • Which numbers will take 6 or more steps? • Can you predict which numbers will take longer than others? • Some people illustrate this process with 4 numbers at the corners of a square, with the difference numbers at their midpoints to form a new square… see next slide

R N Vertes 2020 The 4 digit differences process 2.  (David Cain variation) Choose any four numbers. Place them at the corners of a square. Find the differences between numbers at adjacent corners, mark them halfway between the pairs, and join points to make another square. Continue the process.

R N Vertes 2020 6. Self-descriptive numbers

 Write down a 5 digit number using only 0,1,2,3,4 (but, for ease, not all the same nor exactly one of each)*  The next number in the sequence is, in order, the number of 0s, the number of 1s, the number of 2s, the number of 3s and the number of 4s in your previous number  Repeat. Can you find a number which repeats the previous line? You have found a self-descriptive number

 Try with a 4-digit number using 0,1,2,3  Try with a 10-digit number using 0-9  Try …. *What would happen if ‘all same’ or ‘one of each’?

R N Vertes 2020 7. 4 Digit process: Descending order, reverse, subtract, order, reverse, add

• 1. Take any 4 digit number (not all the same digits) • 2. Put the numbers in descending order. • 3. Subtract, from this number, its reverse • 4. Put this new number in descending order. • 5. To this number add its reverse. • 6. This is your new number.

• 7. Apply steps 2-6 repeatedly. What happens?

• Try this with a 3 digit number. (Try with 5 or more digits....)

R N Vertes 2020 8. Ignore the Tens

• Write down any two single digits and add them together • (Perhaps try 0 and 1 to start ?) • If the result is a two digit number then disregard the tens digit. • Write the result down and then add the last two single digits in the list to create a chain. • For example 3, 6 → 9 → 5 → 4, → … What happens? • Can you account for every possible 2-digit starting combination? • How many should there be? • The next 3 slides, from David Cain, illustrate some interesting patterns if some of the results are displayed in a circle

R N Vertes 2020 Ignore the Tens illustrated 1.

R N Vertes 2020 Ignore the Tens illustrated 2.

R N Vertes 2020 Ignore the Tens illustrated 3.

R N Vertes 2020 9. Digital sum

• Start with any 2 single digit numbers, and write them down as the first two numbers in a sequence. (Perhaps try 0 and 1 first?) • Add the two numbers and find their digital sum (i.e. if more than 9, then add the digits together until you obtain a number between 1 and 9). This is your new term in the sequence, write it down. • Add the last two terms written down, and find their digital sum, to make the next term in the sequence. • For example 2 → 6 → 8 → 5 → 4 → … • Explore what happens over a good number of iterations, with different starting pairs. • Can you account for every possible 2-digit starting combination? • How many should there be? • Can you illustrate the results in circles, as with “Ignore the Tens”?

R N Vertes 2020 10. WORD ITERATION CHOOSE A WORD

WRITE DOWN THE WORD

COUNT THE NUMBER OF LETTERS IN THE LAST WORD YOU WROTE

WRITE THIS NUMBER AS A WORD

ARE THE LAST 2 WORDS WRITTEN EXACTLY THE SAME? NO

YES What word do you get? Try with other starting words. STOP What happens in languages R N Vertes 2020 other than English? 11. NUMBER ITERATION • Try e.g. with 2,5,1 B ≠ 1 • Try other triples Choose A, B,C Try A,B,C as +ve integers

Write down A

Divide A by B, and then add C

• How does the final number relate Write down this answer to A,B.C?

Are the Last two numbers written EXACTLY the same (to 2 d.p.)? NO

YES

Ho STOP R N Vertes 2020 12. Litov’s Mean Value Theorem A. Choose two numbers. (Perhaps best to start with two positive integers). • Generate a sequence when each new term is the (arithmetic) mean of the previous two numbers • Stop when the last two terms are the same to 2 decimal point accuracy • How is your answer related to your two chosen numbers? • Repeat with another two starting numbers • If you choose to start with { a, b } what will be the limit ? • Can you prove the result?

B. What happens if you start with three numbers, and each next term is the arithmetic mean of the previous three? (When using algebra, try to stay in fractions to see some interesting patterns). C. What happens with 4 numbers … ?

R N Vertes 2020 13. Sum of the squares of the digits (also known as Happy Numbers)

• Take any 2 digit number. (e.g. 16) • The next number in the sequence is the sum of the squares of the digits i.e. 10a+b → a² + b² • The sequence continues so each next number is the sum of the squares of the digits in the previous number. You stop when you obtain a previous number in the sequence. • Example: 16 → 37 → 58 → 89 → 145 → (1+16+25) 42 →….. • Apart from 00 (→ 00) are there any other numbers which do not link into the example sequence? How many such numbers are there? What happens with them? • If you end up at 1, you started at a Happy number. Make a list of happy numbers less than 100. What happens with Unhappy numbers? • What happens when you start with three digit numbers?

R N Vertes 2020 14. New Testament Maths 1. • Take any positive integer number (e.g. 126) divisible by 3 • The next number in the sequence is the sum of the cubes of the digits e.g. 100a+10b+c → a³ + b³ + c³ • The sequence continues so each next number is the sum of the cubes of the digits in the previous number. You stop when you obtain a previous number in the sequence. • Example: if choosing 24, then find 2³ + 4³ (= 8 + 64) = 72 • Now find the sum of the cubes of the digits in your new answer e.g. 7³ + 2³ (= 343+ 8) = 351 so 24 →351 → • Continue. What happens?

• Try other numbers divisible by 3. Explain.

R N Vertes 2020 New Testament Maths 2.

St John Ch 21 v11

“Simon Peter went up and drew the net to land full of great fishes, One hundred and fifty three of them: and for all there were so many, yet was not the net broken”

R N Vertes 2020 New Testament Maths 3. Extending the investigation We discovered that 1³ + 5³ + 3³ = 153

How many more numbers like 153 with this property can be found?

e.g. 3³ + 7³ + 0³ = 27 + 343 = 370 so 370 is one such number

Does it matter if you start with numbers divisible by 3 with more than 3 digits?

R N Vertes 2020 15. Sum of the cubes of the digits

• This is generalising the “New Testament Maths” process • Take any 3 digit number (e.g.145) • The next number in the sequence is the sum of the cubes of the digits i.e. 100a+10b+c → a³ + b³ + c³ • The sequence continues so each next number is the sum of the cubes of the digits in the previous number. You stop when you obtain a previous number in the sequence. • Example: 145 → (1+ 64+125) 190 → (1+ 729+0) 730 →….. • Which 1/2/3 digit numbers do not link into the example sequence? How many such numbers are there? What happens with them?

R N Vertes 2020 16. Sum of the fourth powers of the digits

• Take any positive integer • The next number in the sequence is the sum of the fourth powers of the digits • The sequence continues so each next number is the sum of the fourth powers of the digits in the previous number. You stop when you obtain a previous number in the sequence. • E.g: 145 →(1+256+625) 892→(4096+6561+16)10673 →…..

• What interesting patterns can you discover ? • Are there any numbers such that they are the sum of the 4th powers of their digits? • Is this process is programmable? Can you use Excel to do it?

R N Vertes 2020 17. Back Front Shuffles 1.

• 6 people each with a number 1-6 stand in a line in numerical order [1 2 3 4 5 6] • A new row in front of them is made, moving in turn. Each completed row counts as one round. • The person at the back starts a new line, then the person at the front moves, then the next from the back, then the next from the front : result [6 1 5 2 4 3] • Count the number of rounds until you reach the original line-up sequence. • What do you think will happen for other numbers of people? Try for 8 people. How about for 7? 9? 10? 5? • Is there a formula predicting required rounds for N people?

R N Vertes 2020 Back Front Shuffles 2. Tabulating the results • It can help to draw up a table 1 2 3 4 5 6 • Record the starting position 6 1 5 2 4 3 • .. and then the first iteration • You might then complete the table, until the original row recurs, in a number of ways • It can be done by using the last iteration to produce the next by applying the process • OR Notice how each numbered position changes , e.g. 1 was followed into its position by 6; 6 by 3, 3 by 5 … • This can help complete the table by 1 2 3 4 5 6 using it in each column from row to row ; 6 1 5 2 4 3 or, even more quickly, give you a guide 3 to the number of steps to return to the start 5 (See “People Maths Hidden Depths” book) 4 • When do we get “fewer than expected” rounds ….. (fewer than the number of people)?

R N Vertes 2020 18. The Princess chooses

• A princess has to choose a suitor to marry. The King has organised a group of (say) 6 “suitable” people, and says she must choose fairly between them, though she has privately made a choice. She is a mathematician: she decides to make them all stand in a circle, and from where she starts counting, every second one has to drop out. Where does she plant her chosen suitor, so he ends up as the last man standing – and gets to marry her? • You can use People Maths to do this “live”, and use a table as in Back Front Shuffles to track the process • What is her rule for a collection of N people using “every second one drops out” to make her chosen suitor the winner? • What is her rule for positioning the winner if she drops out the 1st, keeps 2nd, drops out 3rd, keeps 4th, etc.? • What is her rule if she drops out every 3rd? (every 4th, every ‘k’ th?) • If you use tables to track the processes, what patterns can you find?

R N Vertes 2020 19. Card Shuffle Sequences Restored 1.

• A set of cards (say 1-6) prearranged in a certain sequence is taken. • The first visible is placed at the back, and the next is displayed and discarded. It’s 1. The next is placed at the back, and the next displayed and discarded – it’s 2; and so on until all the original cards have been ‘discarded’. The set of discards turns out to be in numerical order. • Can you discover in what sequence the set was at the start? What methods can we use, apart from trial and improvement? Can you try via a “People Maths” method? How might you record the results? • Can you create the sequences using the “place at the back/discard” process to produce 1-10? Can you predict its order for numbers 1-100? Ace to King in one suit of a pack of cards? A set of letters A to Z? A normal pack of 52 cards?

R N Vertes 2020 Card Shuffle Sequences Restored 2. • A set of cards (say 1-6) prearranged in a certain sequence is taken. • The person managing counts 1, and the first visible is placed at the back, and the next is displayed and discarded. It’s 1. The person managing counts out two cards, one at a time, placing them at the back, and the next card is displayed and discarded – it’s 2. The person now counts, 1,2,3, always putting a card at the back; the next card is a 3. This process continues, each time counting out one more card than before, until all the original cards have been ‘discarded’. The set of discards turns out to be in numerical order. • Can you discover in what sequence the set was at the start? What methods can we use, apart from trial and improvement? Can you try via a “People Maths” method? How might you record the results? The table offered for Back Front Shuffles could help. • Can you create the sequences using the “place at the back/discard” process to produce 1-10? Ace to King in one suit of a pack of cards? Letters A to Z?? Can you predict its order for numbers 1-100? A pack of 52 cards?

R N Vertes 2020 Card Shuffle Sequences Restored 3. • A set of cards (say 1-6) prearranged in a certain sequence is taken. • The person managing the process spells out ONE a letter at a time, and for each letter moves the card to the back. The next is displayed and discarded. It’s 1. Now TWO is spelled out, each time a letter is used it is moved to the back. The next card is displayed and discarded – it’s 2. Now THREE is spelled out and the next card visible is 3, and so on until all the original cards have been ‘discarded’. The set of discards turns out to be in numerical order. • Can you discover in what sequence the set was at the start? What methods can we use, apart from trial and improvement? Can you try via a “People Maths” method? How might you record the results? The table offered for Back Front Shuffles could help. • Can you create the original sequence using ten cards 1-10, and the above counting process, to produce 1-10 in sequence?

R N Vertes 2020 20. Circle Patterns • Draw a circle. A good size is radius 5 cm. • Choose any point on the circumference and join it to another point. Mark on the sheet the direction of the line 1. • From your previous arrival point, draw a line to another point on the circumference. Mark on the sheet the direction of the line 2. • From your previous arrival point, draw a line to another point on the circumference. Mark on the sheet the direction of the line 3. • Repeat drawing lines, in turn parallel to lines 1,2,3 from each previous arrival point. Using a ruler to ensure lines are parallel will help. • What (usually) happens? Can you explain why? • Try again with a circle and 3 new directions of lines. • What happens if you start with a square? An ellipse? (Other polygons? Other numbers of lines?)

R N Vertes 2020 21. Spirals

• Take a sheet of A4 squared paper, best if 5mm sides of squares. • Choose three digits 1-9 in any order. Do not make all three the same. • In the middle of the page mark a dot at an intersection. • If your three digits are A, B, C, then from the initial dot, draw a line A units to the East/right/x direction. From where this finishes turn through 90º anticlockwise and draw a line B units North/in y direction. From where this finishes turn through 90º anticlockwise and draw a line C units West/in (- x) direction. From where this finishes turn through 90º anticlockwise and draw a line A units South/in (-y) direction. From where this finishes turn through 90º anticlockwise and draw a line B units. From where this finishes turn through 90º anticlockwise and draw a line C units. Keep going, turning through 90º anticlockwise and drawing a line successively A,B,C units until… (what happens?) • Explore the geometric designs created; what different types can you classify? Area? • Try with the original 3 numbers in a different order; 4 or more numbers; your landline or mobile phone number, etc. {This idea has been around for a while as “Spirolaterals”] • Try using isometric paper and 120º turns (with 3 or 4 numbers)

R N Vertes 2020 22. Mathematical Origami: - How to trisect a sheet of paper using trial and improvement

Start with a piece of paper roughly the same width and length as a 30 cm ruler.

• Guess one third from one end, mark it. Your guess can be ‘wild’! • Fold the other end to your guess and mark its fold line. Open out the paper. • Fold, mark, open out alternate ends to the last of the marked fold lines, repeatedly. You will know when to stop!

R N Vertes 2020 23. Mathematical Origami: - Follow-up:

• Dividing a length of paper into 2, or 4 or 8 equal parts is ‘easy’:- each can be folded exactly by halving. • We can now use what we learned via trisection to divide a length of paper into 6 equal parts. To make 9 follows… • Use what you have learned in the trisection process to divide a sheet of paper into 5 equal parts. • Can you adapt the process to divide the length into 7 equal parts?

• Going back to the trisection, can you explore why the method works by looking at the errors and how they diminish?

R N Vertes 2020 24. Persistent numbers  Start with a 2 digit number. This is round count 0.  Obtain a new number by multiplying the 2 digits together. Add one to the round count each time you do such a multiplication.  When you get a single digit treat it as a two digit number with a zero in the tens place. Repeat until you obtain 0. The number of rounds to achieve this is the persistence of the original number.

 Which 2 digit numbers have the greatest persistence? (It can be useful to use a 100 square grid to record results; and it is great if you can use a 1-100 multilink board, building towers of “persistence height” with coloured cubes to get a 3-D perspective). A Tens and Ones table is given on the next slide

 Which 3 digit numbers have the greatest persistence, taking the product results as a three digit number? How can we record?

R N Vertes 2020 Tabulating the results: a Tens and Ones table

ONES

Rule 0 1 2 3 4 5 6 7 8 9

0

1

2

T 3

E 4 N S 5 6

7

8

9

R N Vertes 2020 25. Molecules

• Gather into a group of (recommended size) 10 people. • Split into a number of separate groups. Some of these can be the same size. Count the numbers in each group and record the group sizes. • At each round one person from each team leaves and, with the other leavers, forms a new group. Count the numbers in each group and record the new group sizes. • Continue the process until …. • Try with 15 people; try with 7 or 8 people; predict... • When do (mathematically) ‘nice’ things happen?

R N Vertes 2020 26. Tens added to twice Ones 1. •. Take any 2 digit number (e.g. 15) • The next number in the sequence is the first digit added to double the second digit i.e. 10a+b → a+2b • The sequence continues this way: i.e. each next number uses the previous one, finding the sum of the 1st digit and double the 2nd • You stop when you obtain a previous number in the sequence • Example: 15 → 11 → 03 → 06 → 12 → 05 → 10 → 01 →….

• Apart from 00 (→ 00) there are some other numbers which do not link into the loop in the example. How many such numbers are there? What connects them?

R N Vertes 2020 Tens added to twice Ones 2.

R N Vertes 2020 69 88 29 48 67 86 43 62 24 1 20 96 77 58 10 2 31 81 21 39 98 79 50 5 4 40 97 44 25 23 42 78 59 63 12 8 61 82 95 80 87 22 84 65 68 41 6 16 46 27 99 49 60 76 19 38 30 3 13 92 73 34 57 35 54 53 11 7 70 72 91 32 51 36 55 15 14 83 64 74 93 90 26 37 17 9 45 18 33 56 94 28 85 71 75 52 47 66 89 27. DOTS –Difference of two squares

• Take any 2 digit number. (e.g. 15) • The next number in the sequence is the difference between the square of the first digit and the square of the second digit i.e. 10a+b → |a² - b² | (taking the positive value of the difference) • The sequence continues so each next number uses the previous one, finding the difference of the squares of the 1st and 2nd digits • You stop when you obtain a previous number in the sequence • Example: 15→ 24 → 12 → 03 → 09 → 81 → 63 → 27 →….. • It might help to use a Tens and Ones table to record what becomes of each number • Apart from 00 (→ 00) are there any other numbers which do not link into the example sequence? How many such numbers and sequences are there?

R N Vertes 2020 28. Another 2 digit repeated rule

• Take any 2 digit number (e.g. 15) • The next number in the sequence is the first digit added to the square of the second digit i.e. 10a+b → a + b² • The sequence continues so each next number uses the previous one, finding the sum of the 1st digit and the square of the 2nd digit. • You stop when you obtain a previous number in the sequence. • Example: 15 → 26 → 38 → 67 → 55 → 30 → 03 → 09 →….. • It might help to use a Tens and Ones table to record what becomes of each number • What happens? Apart from 00 (→ 00) are there any numbers which do not fit into the example sequence? What happens to them? How many such sequences are there?

R N Vertes 2020 29. Yet another 2 digit repeated rule • Take any 2 digit number • The next number in the sequence is the first digit added to 4 times the second digit i.e. 10a+b → a+4b • You stop when you obtain a previous number in the sequence • Example: • 15 → 21 → 06 → 24 → 18 → 33 → • It might help to use a Tens and Ones table to record what becomes of each number • What patterns can you find? Find out what happens to every two digit number. Apart from 00 (→ 00) there are some other numbers which link to themselves. How can you find them? • Explore other variations e.g. the first digit added to 3 (5,6,7,8,9) times the second digit i.e. 10a+b → a+Nb

R N Vertes 2020 30. More 2 digit repeated rules • Take any 2 digit number • Explore other variations e.g. the first digit added to 3 (5,6,7,8,9) times the second digit i.e. 10a+b → a+Nb • You stop when you obtain a previous number in the sequence • Example: (for a+3b) • 15 → 16 → 19 → 28 → 26 → 20 → 02→ 06 → 18 → …. • What patterns can you find? Find out what happens to every two digit number. Apart from 00 (→ 00) there are other numbers in some of the “a+Nb” sequences which link to themselves. Can you find them? • It might help to use a Tens and Ones table to record what becomes of each number • Explore other variations e.g. the first digit added to 3 (5,6,7,8,9) times the second digit i.e. 10a+b → a+Nb

R N Vertes 2020 Chains - Shuffles, Loops and Fixed Points Repeated Instructions and Processes Leading to Interesting Results by Bob Vertes

A collection of activities suitable for KS2/3/4 and older. Many were shared by the author at ATM Easter conferences and at some branch meetings. Previously they were explored in secondary classrooms with student teachers and during INSET with practising teachers.

Some of the activities have been adapted from work shared by David Cain, who during his lifetime was an active member of the ATM. Most activities are repeated processes (iteration) with positive integers, usually reaching a fixed point after a number of steps, or a loop. Sometimes more than one fixed point, or more than one loop, occurs.

The question not to forget to ask is ” Why?”

• Activities 8, 9, 10, 20, 21, 24, 25 are for KS2 or older • Activities 1-7, 13, 17-19, 22, 23 are for KS3 or older • Activities 26-30 are suited for KS4 • Activities 11,12,14-16, are for KS4/5, best with a calculator, Excel or via a computer program.

Association of Teachers of Mathematics 2A Vernon Street, Vernon House, Derby DE1 1FR Tel: 01332 977891

Email: [email protected] www.atm.org.uk