Enigma 209: Double number slab
25 July 2014
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From New Scientist #1355, 28th April 1983 [link]
A double number slab is just two rows of squares in a rectangle, with the correct number in each square. The numbers in the top row are just 1, 2, 3, …, n. In each bottom row square is written the number of times the number above it occurs in the completed slab.
So, for instance, if n=5, you fill in the top row as shown, and in the bottom row you replace A by the number of 1’s, B by the number of 2’s, …, E by the number of 5’s in the whole slab.
Given that n is at least 4, can you say:
(a) for what value of n is it impossible to complete the slab properly?
(b) for what value of n is each second-row number less than or equal to every number to the left of it?