A MATHEMAGICAL MYSTERY TOUR 3: … A RECURRING RELATION? MIKE SMITH MATHEMATICAL ASSOCIATION: STIRLING CONFERENCE OCTOBER 2019 [email protected] SIMÉON POISSON "LIFE IS GOOD FOR ONLY TWO THINGS, DISCOVERING MATHEMATICS AND TEACHING MATHEMATICS" LEWIS CARROLL AKA CHARLES LUTWIDGE DODGSON “BEGIN AT THE BEGINNING," THE KING SAID, VERY GRAVELY, "AND GO ON TILL YOU COME TO THE END: THEN STOP.” • LAWN FOR LUNCH! OR ... IT WASN’T THE GARDEN’S FAULT ALICE’S • THE FARMER DIDN’T HAVE TIME TO TRIM THE GRASS HIMSELF – SO HE PUZZLES LEFT THE JOB TO HIS LIVESTOCK … … • HISOW C AND HIS GOAT WOULD EAT ALL THE GRASS IN 45 DAYS • THEOW C AND THE GOOSE WOULD EAT ALL THE GRASS IN 60 DAYS THROUGH • AND THE GOAT AND THE GOOSE WOULD EAT IT ALL IN 90 DAYS THE • ASSUMING THE GRASS IS NO LONGER GROWING, IF THE FARMER LEFT THE COW, GOAT AND GOOSE IN THE GARDEN TOGETHER HOW LOOKING LONG WOULD IT TAKE TO EAT ALL THE GRASS? GLASS • LAWN FOR LUNCH! OR ... IT WASN’T THE GARDEN’S FAULT • THE ANIMALS WILL EAT ALL THE GRASS IN 40 DAYS! • WHY? 1 8 • COW + GOAT = 45 = 360 1 6 COW + GOOSE = = • 60 360 ALICE’S PUZZLES 1 4 … … • GOAT + GOOSE = = THROUGH THE 90 360 LOOKING GLASS • LAWN FOR LUNCH! OR ... IT WASN’T THE GARDEN’S FAULT 14 • 2 COWS + GOAT + GOOSE = (8 + 6) 360 • SO 2 COWS = 10 (14 – 4) 360 5 • SO 1 COW = WHICH MEANS … 360 3 1 • 1 GOAT = AND 1 GOOSE = 360 360 9 1 ALICE’S PUZZLES • SO COW + GOAT + GOOSE EATS IN A DAY ( ) … … 360 40 THROUGH THE LOOKING GLASS • SO 40 DAYS TO EAT ALL THE GRASS A MATHS TEACHER NEVER SWITCHES OFF! ANTONI GAUDI • “I AM A GEOMETRICIAN, MEANING I SYNTHESISE.” SAGRADA FAMILIA • THOUGHT I WAS GOING TO SPEND A COUPLE OF HOURS LOOKING AT STAINED GLASS WINDOWS! … … … • [OH! GAUDI I SAID TO MYSELF!] • BUT NO! SOON DISCOVERED A HUGE AMOUNT OF MATHEMATICS IN THE CATHEDRAL • GEOMETRY OF THE PLACE – EVERYTHING IN MULTIPLES OF 7.5 • USE OF CONICS; HYPERBOLOIDS, PARABOLOIDS, HELIOCOIDS, ELLIPSOIDS, TWISTED COLUMNS • OF COURSE, THIS APPLICATION OF MATHEMATICS DID NOT EXCITE ME AS MUCH AS … … THE MAGIC SQUARE ON THE PASSION FACADE TOOK A WALK OUTSIDE AND WHAT DID I SEE? … … MAGIC SQUARE • AS YOU KNOW A 4X4 MAGIC SQUARE, USING 1-16 WOULD HAVE A MAGIC CONSTANT OF 34 • IE ROWS, COLUMNS, DIAGONALS WOULD ADD TO 34 • LOOK CLOSELY AT THIS ONE • THE 12 AND 16 HAVE BEEN REPLACED BY 10 AND 14 • THE MAGIC CONSTANT IS 33! • WHY CHANGE? • SUBIRACHS TOOK AN EXISTING MAGIC SQUARE AND ADAPTED IT. WHY? • THE MAGIC CONSTANT OF 33 IS THE AGE JESUS IS TRADITIONALLY BELIEVED TO HAVE BEEN EXECUTED • BUT THIS IS NOT AN ‘ORDINARY’ MAGIC SQUARE IN MORE WAYS THAN THAT • CHECK THAT ALL ROWS, COLUMNS AND DIAGONALS ADD UP TO 33 … … YUP! MAGIC • BUT HOW ABOUT THESE … … SQUARE There are over 300 combinations which add up to 33 And one last thought … … AND NOW FOR SOMETHING COMPLETELY DIFFERENT! FUN WITH 79? • 7+9 + 7X9= 79 [SUM + PRODUCT = NUMBER] • SMALLEST PRIME FOR WHICH THIS WORKS • EMIRP [REVERSIBLE PRIME; 79 … 97] • AN EMIRP WHICH IS THE SUM OF 3 OTHER EMIRPS • 79 = 11 + 31 + 37 [11+13+73 = 97] • SMALLEST PRIME WHOSE SUM OF DIGITS IS A 4TH POWER [16=24] • TRY 27 – 72 [YUP! 79] • IT IS A “HAPPY PRIME” ITERATION OF 72 + 92 = ENDS AT 1 • SUBTRACT 79 FROM 7! + 9! AND GET ANOTHER EMIRP! ON A LIGHTER NOTE … • 1979 SAW THE FIRST EPISODE OF STAR TREK • AD79 GREATEST ERUPTION OF MOUNT VESUVIUS • AROUND THE WORLD IN 80 DAYS PHINEAS FOGG DID IT IN 79 DAYS • 79 IS A COUSIN PRIME TO 83 • [DIFFER BY 4 AS OPPOSED TO TWIN PRIMES (2) AND SEXY PRIMES (6) 79 IS ALSO A SEXY PRIME [WITH 73] TH • THE 79 TRIANGULAR NUMBER IS 3160 • WHICH IS THE FIRST TO CONTAIN ALL TRIANGULAR NUMBERS IN ITS DIGITS • THE ATOMIC NUMBER OF CHEMICAL ELEMENT GOLD (AU) IS 79 IT WOULDN’T BE MATHEMAGICAL WITHOUT Π WWW.MYPIDAY.COM 260256 LIGHTENING MULTIPLICATION • ASK A STUDENT TO GIVE YOU A 3-DIGIT NUMBER • SAY 567 WRITE IT DOWN TWICE 567 567 • ASK FOR ANOTHER, SAY 382 WRITE IT DOWN UNDER LEFT 382 617 • NOW YOU WRITE ANOTHER NUMBER UNDER 2ND 567, SAY, 617 • NOW SAY YOU ARE GOING TO DO THE TWO MULTIPLICATIONS AND ADD THEM • IN YOUR HEAD! • AMAZE THEM WITH YOUR ANSWER OF 566 433 • CAN YOU SEE HOW THIS WORKS? LIGHTENING MULTIPLICATION • YOU WRITE DOWN 567 567 • THEY GIVE ANOTHER NUMBER 382 • YOU WRITE DOWN THE “9 COMPLEMENT” 617 • YOU GET THE SUM OF PRODUCTS BY; • SUBTRACT 1 FROM MULTIPLICAND … IN THIS CASE 567 – 1 IS 566 • THE 9 COMPLEMENT OF 566 IS 433 • SO THE SUM OF THE PRODUCTS IS 566 433 LIGHTENING MULTIPLICATION • HOW DOES IT WORK? • IF YOU THINK WHAT YOU ARE DOING IT IS 567 X (382 + 617) • WHICH IS 567 X 999 • WHICH IS 567 X 1000 – 567 • WHICH IS 566 000 + ‘9 COMPLEMENT’ • IE 566 433 A NICE LITTLE CODE HOW MUCH CHANGE DO YOU HAVE IN YOUR POCKET? • A NICE LITTLE TRICK TO DRAW THE STUDENTS IN AT START OF PERIOD • ASK THEM TO COUNT CHANGE IN POCKET (BEST TO WORK IN PENCE NOT £S AND P – ALTHOUGH IT WILL WORK FOR £P) • PROBABLY BEST TO WRITE DOWN! • MULTIPLY BY 2 • ADD 3 • MULTIPLY BY 5 • SUBTRACT 6 • GET VOLUNTEER TO TELL YOU FINAL NUMBER (EG 639) • YOU TELL HIM/HER S/HE HAD 63 PENCE IN POCKET! • (LAST DIGIT WILL ALWAYS BE 9 – SCORE OUT!) HOW MUCH CHANGE DO YOU HAVE IN YOUR POCKET? • HOW DOES IT WORK? • START WITH A • MULTIPLY BY 2 GIVES 2A • ADD 3 GIVES 2A + 3 • MULTIPLY BY 5 GIVES 5(2A + 3) = 10A + 15 • SUBTRACT 6 GIVES 10A + 9 • SO WHATEVER NUMBER YOU STARTED WITH HAS BEEN MULTIPLIED BY 10 AND THE LAST DIGIT WILL ALWAYS BE 9 NUMBER CHAINS • ANOTHER NICE ONE – WHICH CAN EXPAND AS MUCH AS YOU WANT! • TAKE A TWO DIGIT NUMBER, SAY 23. • MULTIPLY UNITS DIGIT BY 4 AND ADD TENS DIGIT – WHAT HAPPENS? 23 147 1 18 33 15 35 38 29 24 6 21 • CAN YOU FIND A NUMBER WHICH FORMS A “ONE LINK” CHAIN • TRY 13 (OR 39) NUMBER CHAINS • TRY MULTIPLYING UNITS DIGIT BY 3 AND ADD TENS • REPEAT USING X2, OR X5 • TRY TO FIND SINGLE CHAIN STARTING NUMBER- CAN YOU SEE A PATTERN?. • EG X2 + T • 23, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8 • SINGLE LINK NUMBER IS 19 • EXPERIMENT WITH X3 + T, X5 + T, X6 + T ETC NUMBER CHAINS X unit + tens Single link start number X2 + t 19 9x2+1=19 X3 + t 29 9x3+2=29 X4 + t 39 9x4+3=39 X5 + t 49 X6 + t 59 Any multiple of the number will also take you back to single link number Eg x2 + 1 … … start at 38. 8x2+3 = 19. start at 57, 7x2+5 = 19 Again, can you look at the algebra behind this? What we have is 9n + (n-1) = 10n – 1 so 10 x 2 – 1 = 19 10 x 3 – 1 = 29 etc MOBIUS STRIP • SQUARES FROM CIRCLES? • A VALENTINE’S DAY MOBIUS STRIP! Japanese methdod MULTIPLYING Egyptian Gelosia method method (Napier’s bones) JAPANESE METHOD 23 x 12 = (2x10 + 3)(1x10 + 2) = 2x1x102 + [2x2x10 + 3x1x10] + 3x2 = 276 GELOSIA OR NAPIER’S BONES NAPIER’S BONES (OR RODS) EGYPTIAN METHOD 23 x 12 1.. 12 2.. 24 4.. 48 16+4+2+1 = 23 8.. 96 Add up corresponding values 16.. 192 32.. 384 192+48+24+12 = 276 HAILSTONE NUMBERS Start with any number. If even divide by 2 If odd x3 + 1 … … … keep on going .. … what happens? 52 40 34 22 26 20 16 10 8 14 17 13 4 11 5 2 7 1 TRAINS AND A FLY 2 trains are on the same track 100 miles apart. They are heading towards each other at 50mph A fly leaves the front of one train, flying at 60mph towards the other train When it reaches that train it immediately turns around and flies back The fly repeats this until the trains crash – squishing it completely! How far had the fly travelled in total? TRAINS AND A FLY You can get caught up in summing a series! Easier to think it through! … Trains 100 miles apart, travelling at 50mph will take 1 hour to meet Fly is moving at 60mph – so will travel 60 miles before getting squashed! AND FINALLY … … Can you insert the numbers 1- 8 on this cube to make all the faces add up to the same total? A start has been made AND FINALLY … … THANK YOU FOR BEARING WITH ME! ENGLISH IS I M P O R T A N T BUT MATHEMATICS IS I M P O R T A N T E R ! SOME EXTRAS! MULTIPLYING BY 11! 11 X 34 = 3 4 (3+4 = 7) = 3 7 4 = 374 11 X 78 = 7 8 (7+8 = 15) = 7 15 8 = 8 5 8 = 858 HAPPY NUMBERS 31 IS A HAPPY NUMBER! WHY? 32 + 12 9 + 1 =10 12 + 02 1 + 0 = 1 IF END UP AT 1 => HAPPY NUMBER! PERFECT NUMBERS 6 IS A PERFECT NUMBER WHY? FACTORS OF 6 (NOT INCLUDING 6) 1, 2, 3 1+2+3 = 6 NEXT PERFECT NUMBER IS 28; 1+2+4+7+14 = 28 WHAT ARE THE NEXT PERFECT NUMBERS? DO YOU SEE ANY PATTERN? NEXT FEW PERFECT NUMBERS • 6 (EUCLID) • 28 (“) • 496 (“) • 8 128 (“) • 33 550 336 (UNKNOWN CIRCA 1456) • 8 589 869 056 (CATALDI 1588) • 137 438 691 328 (“) • 2 305 843 008 139 952 128 (EULER 1772) • PATTERN? SO FAR ALL END IN EITHER 6 OR 8.