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#9 Rectangulator

Set by Katie Steckles

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 the first button you press is the 7 key, how many possible four-digit numbers can you create?

What about if your square or rectangle is allowed to have a height or width of zero?

Can you show that for any square or rectangle you choose, the resulting four-digit number will always be divisible by 11?

Solution next week

#8 Two Cubes

Solution

The points are √2 apart (using Pythagoras’s theorem, the length of the diagonal is the square root of 1² + 1² = 2). We can find pairs of points for all values of n from 1 to 6 by the same logic. The square root of 3 (1² + (√2)²) is the length between opposite corners of one cube. The long edge of the frame has a length of √4 (or 2), while √5 and √6 are the diagonals of the long face and whole frame, respectively. Sadly, a frame with a third cube added on the end won’t get you √7 or √8, but it will contain √9, √10 and √11.