**From New Scientist #2474, 20th November 2004**

I gave a copy each to my niece and nephew of the following grid and instructions to see what different answers they might produce.

“Follow these instructions, in each case writing a number, larger than 10, across or down, within the grid, and with one digit in each box:

A. Starting at a box number which is a square, write a square.

B. Starting at a box number which is a cube, write a cube.

C. Starting at a box number which is a prime, write a prime whose digits add up to an even number.

D. Starting at a box number which is even, write an even number.

Your answers should fill the grid, with no two answers overlapping.”

Unfortunately, one of the copies had instruction D completely missing. Each child gave me a 3-by-3 grid which was correct for the copy they had been given. By coincidence they gave me identical grids.

What was their completed grid?

**Enigma 1772** is also called “Clueless”.

[enigma1316]

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This direct approach, programmed in Python 3, runs in 5.1s. With a bit of analysis we can do better than this, but I like this approach as it doesn’t need to make assumptions to solve the problem.

Solution:The completed grid is shown below:The two different ways of filling out the grid are shown below:

A: (red) A square, starting from a box numbered with a square.

B: (green) A cube, starting from a box numbered with a cube.

C: (blue) A prime with an even digit sum, starting from a box numbered with a prime.

D: (yellow) An even number, starting from a box numbered with an even number.