December 30th, 2024

A Puzzle about a Calculator

Katie Steckles discusses her first year managing the BrainTwisters column, presenting a puzzle about forming four-digit numbers from a calculator keypad, all of which are divisible by 11.

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Katie Steckles reflects on her first year managing the puzzle column at New Scientist, now called BrainTwisters, by presenting a mathematical puzzle involving a standard calculator. The challenge is to press four digit keys that form the corners of a square or rectangle on the keypad, starting with the 7 key, and to determine how many four-digit numbers can be created. The solution reveals that eight numbers can be formed with a height or width of 1 or 2, and additional numbers can be created if zero height or width is allowed. Notably, all resulting four-digit numbers are divisible by 11. The explanation involves algebraic manipulation to show that the sum of the digits in the constructed numbers maintains divisibility by 11. An alternative proof is provided through examining the distances of each digit from the nearest multiple of 11, demonstrating that the alternating sum of the digits equals zero. The column continues to explore intriguing mathematical concepts, encouraging readers to engage with the puzzles.

- The puzzle involves creating four-digit numbers from a calculator keypad.

- All resulting numbers from the puzzle are divisible by 11.

- The solution includes both algebraic and alternative proofs for divisibility.

- The column aims to engage readers with interesting mathematical challenges.

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7 comments

By @qrian - 3 months

Any parellogram would work since given four digit number n1n2n3n4, it is divisible by 11 iff n1+n3=n2+n4, and each ni is linear combination of the coords of keypads xi, yi, and thus (n1+n3)/2 = (n2+n4)/2

By @gorkish - 3 months

For a 4 by 4 hexadecimal keypad with 0x0..0xF keys in any sequential row or column arrangement the same will be true that all possible numbers will be a multiple of 0x11.

This holds true for any size square keypad; the common multiple will be 1+x^2 and for square-number bases, the value of the common factor will always be “11” in conventional symbology. A 2x2 base 4 and even a 1x1 binary keypad respects the rule although it’s sort of meaningless in the latter context.

By @teucris - 3 months

At first I thought this was going to be a puzzle about getting to a specific number using certain rules for navigating the pad, including the operation buttons. For instance, by pressing one or two buttons in each row from top to bottom, can you get the calculator to display 70?

By @russdill - 3 months

Weird, every time I try, I just get "A suffusion of yellow"

By @lilyball - 3 months

Interestingly a 45º rotated rectangle using the keys 4 8 6 2 also is divisible by 11. This isn't directly addressed in the solution, although if you change "move both numbers horizontally or vertically by the same distance" to say "and" instead of "or" then it does.

By @jahbrewski - 3 months

How do you "read" an article like this? I would need to pull out some paper, run calculations, etc. to understand this (but perhaps I'm not the intended audience, as a non-mathematician?) - Or is that how you all approach an article like this?

By @d_tr - 3 months

> If you allow zero width/height, you can also have 7777, 7887, 7997, 7447 and 7117.

Why not 7227, 7337, 7557, 7667 too?

The theorem holds for these as well.