Multiplication cube
The net shown is folded up to form a cube. What is the largest possible vertex product?
Age
11 to 14
Challenge level
Problem
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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
Student Solutions
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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Image
The 6 touches the 4 and the 5, as shown.
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Image
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.