| 1. Can you find a subset of these numbers that added together total 100? |
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| 2. Imagine a cube standing on one vertex, with the opposite vertex vertically above it. Now imagine a horizontal plane moving down through the cube. Describe what happens to the intersection of the plane and the cube as the plane moves from the top vertex to the bottom vertex. |
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3. Show that 5n+1 + 62n-1 is divisible by 31 for all positive integer values of n.
4. There is a famous story of the English mathematician, Hardy, visiting the Indian mathematical genius, Ramanujan, in hospital. Hardy remarked that the taxi he had come in had the number 1729, which struck him as not a very interesting number. Ramanujan disagreed, pointing out that it is the smallest positive integer that can be written in two different ways as the sum of two cubes (of positive whole numbers). Is there a smaller positive integer that can be written in two different ways as the sum of two cubes of integers (positive or negative)?
