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#62 Particular patterns in piles

Set by Peter Rowlett

Arrange balls into a row of piles according to these rules:

1. The first and last piles contain one ball.

2. If two neighbouring piles aren鈥檛 the same size, the change in height is either an increase or decrease of one ball.

There are two valid ways to arrange four balls:

How many ways are there to arrange five balls?

How about six balls?

How many ways are there to arrange nine balls?

Solution next week

#61 Mark to mark

Solution

Adding a mark at 5 allows you to measure all possible lengths: 6-5, 2-0, 5-2, 6-2, 5-0, 6-0.

On the ruler marked at 1, for which we can include the ends as points 0 and 11, adding marks at 4 and 9 lets you measure any length but 6 (1-0, 11-9, 4-1, 4-0, 9-4, [not possible], 11-4, 9-1, 9-0, 11-1, 11-0).

Making marks at 2, 7, and 8 lets you measure every length except 10 (8-7, 2-0, 11-8, 11-7, 7-2, 8-2, 7-0, 8-0, 11-2, [not possible], 11-0). Or, marks at 3, 4 and 9 would give the same solution (the ruler is mirrored).