The minimum number of moves needed to solve a Rubik’s cube

2002

YEAR BOOK

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INSTITUTE OF EDUCATION, LEESON STREET, DUBLIN 2

CIARAN MCNAMEE, LOUISE SULLIVAN & EDWARD NAUGHTON

The minimum number of moves needed to solve a Rubik’s cube

Ciaran McNamee, Louise Sullivan & Edward Naughton at the Exhibition.

In our project we determined that the minimum number of moves needed to solve a Rubik’s cube is 20 (or less depending upon the starting state). We initially conducted research along different branches of mathematics before eventually settling on the idea of considering the cube as a series of intersecting planes. After defining the various parts of the cube, we demonstrated that some form of ‘relative positioning’ was needed here (i.e. the idea that blocks only move relative to each other – obvious but it had to be done).

Next we showed that, if a move is made on the cube through a plane (say plane ‘x’) no amount of moves through the other two planes can restore the cube to order. This was like a small, independent proof, being needed later. We broke the rest of the proof into three sections, where movement is permitted in one, two and all planes. In each section, instead of proving that the cube could be solved in a given number of moves, we proved that any position on the cube could reach any other position on the cube in that number of moves. This allowed us to use information obtained in one section more easily in the next. It also meant that we didn’t have to define maximum disorder: we were instead able to show that maximum disorder could be reached in a certain number of moves.

We calculated that, when movement is permitted in a single plane, any position can be reached in two moves. Using this information and the fact that planes relatively have one dimension in common with each other and one dimension which is unique, we were able to show that, when movement is permitted in two planes, the cube can be solved in eight moves. Using a combination on both of these facts, we proved that, when movement is permitted in all three planes, the cube can be solved in 20 moves.

Our next step, which gave us our final formula, was to come up with a general solution (for cubes bigger or smaller than the Rubik’s cube). Our final formula for the number of moves needed to solve a cube is 10(n–1), where n is the number of blocks per side of the cube.

Ciaran McNamee, Louise Sullivan & Edward Naughton, who entered their project in the Senior Group Section in the Chemistry, Physical & Mathematical Sciences Category at the Esat Young Scientist & Technology Exhibition in January 2002, won one of the top prizes – The Best Group Award. Their teacher was Dr Aidan Seery.

This article was sponsored by Samton Limited.