A Puzzle about a Calculator
It’s now been a year since I took over the puzzle column at New Scientist and turned it into the BrainTwisters column. By way of celebration, I thought I’d write up an interesting bit of maths behind one of the puzzles, which I made a note of at the time and have been meaning to share.
Given a standard calculator keyboard, press – in order, going either clockwise or anticlockwise – four digit keys that form the corners of a square or rectangle on the keypad. This will create a four-digit number, e.g. 7469.
If you’d like to take some time to think about this, go away and do that now – then I’ll share the solution.
From the 7, we can go round a rectangle, with width or height 1 or 2, in either direction. Eight numbers are possible: 7854, 7821, 7964, 7931, 7458, 7469, 7128 and 7139. If you allow zero width/height, you can also have 7777, 7887, 7997, 7447 and 7117.
For zero-height or width rectangles, the first two digits are the last two digits reversed. This will always be a multiple of 11, since it will be a sum of multiples of 110 and 1001, both multiples of 11. If we extend the rectangle, we move both numbers horizontally or vertically by the same distance. Instead of 7117, we could extend to the right and get 7128. This won’t affect whether it is a multiple of 11, as we are adding or taking away 11 per column we move. Likewise, extending a zero-height rectangle horizontally, we just add or subtract 33 per row we move.
