# Multiplication Cube

The net shown is folded up to form a cube. What is the largest possible vertex product?

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The numbers on the three faces meeting at any

vertex can be multiplied together.

What is the largest such product for the vertices of the cube?

*This problem is taken from the World Mathematics Championships*

The 4, 5 and 6 will be involved in the largest possible products, so we can begin by seeing whether they touch.

The 4 and the 5 do not touch, as shown in the diagram below, which shows part of the folded net.

The 4 and the 5 do not touch, as shown in the diagram below, which shows part of the folded net.

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The 6 touches the 4 and the 5, as shown.

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This means that the greatest product must include the 6 and the 5 or the 6 and the 4.

Clearly it will include the 3, not the 1, and so it will be 6$\times$5$\times$3 = 90.

Clearly it will include the 3, not the 1, and so it will be 6$\times$5$\times$3 = 90.