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#16 Order, order!

Set by Katie Steckles

Arrange the digits 1-9 in a line so that each pair of adjacent digits differs by either 2 or 3.

Arrange the digits 1-9 in a line so that each pair of adjacent digits sums to a prime number.

Now arrange the digits 1-9 in a line so that each pair of adjacent digits (when read as a two-digit number) appears in the times tables from 1 × 1 up to 9 × 9.

Solution next week

#15 Domino strips

Solution

There are five ways to cover a 4 × 2 rectangle and 13 ways to cover a 6 × 2 rectangle.

Any n × 2 rectangle can be covered using all the same ways you found to cover an (n – 1) × 2 rectangle, you simply add a vertical domino to cover the additional 1 × 2 column. It can also be covered using all the same ways you found to cover an (n – 2) × 2 rectangle, just with a pair of horizontal dominoes to cover the two new 1 × 2 columns. So, the number of ways to cover any n × 2 rectangle is the sum of the ways to cover an (n – 1) × 2 rectangle and an (n – 2) × 2 rectangle.

Because there is one way to cover a 1 × 2 rectangle and two ways to cover a 2 × 2 rectangle, this makes Fibonacci numbers: 1, 2, 3, 5, 8, 13 and so on.