Maths prize for 15 September 2015: a Conway puzzle

JohnHortonConway1987

The problem: Assemble six 2×2×1 blocks and three 1×1×1 blocks into a 3×3×3 cube.

Seven students solved this problem, and could explain how by word of mouth: Hamse, Khaleah, Jetmir, Mugisha, Lou-Lou, Matthew, and Lola.

Mugisha went one better and wrote out the maths of how to solve the problem, so he gets an extra prize.

Solution: Each face contains an odd number of blocks (nine). Each 1×2×2 unit can contribute only an even number of blocks. Therefore, each face and each “layer” must contain one 1×1×1 unit.

That can be done only if two 1×1×1 blocks are at diagonally opposite corners.

The three planes made by cutting across the middle of the 3×3×3 cube lengthways, widthways, and depthways all contain an odd number of blocks (nine). Therefore, each of those three planes must contain the third 1×1×1 unit. It must be right in the centre of the whole cube.

Once you have placed the three 1×1×1 unit, it is easy to place the 1×2×2 blocks.

Here’s the solution. In the picture all the 1×1×1 blocks are shown as white and all the 1×2×2 blocks as coloured.

Slothouber-Graatsma_Solution

And here’s the key to the solution of the harder 5×5×5 cube problem – assemble 13 1×2×4 blocks, one 1×2×2 block, 3 1×1×3 blocks, and one 2×2×2 block into a 5×5×5 cube – and the full solution.

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John Conway was a lecturer at Cambridge when I studied there. He wasn’t yet famous then. He lectured on number theory. I went for a different option, because I thought number theory seemed a quaint sidetrack. I got to know Conway only when other students and I published a criticism of the maths syllabus, called big meetings to discuss it, and pushed the professors into setting up a committee to reform the syllabus with elected student members. I was one of the elected students, and Conway was one of the lecturers appointed to the committee. The lesson of the whole thing is that I should have thought more before dismissing number theory so stupidly.