36 lines
1.5 KiB
Markdown
36 lines
1.5 KiB
Markdown
Each of the six faces on a cube has a different digit (0 to 9) written on it;
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the same is done to a second cube.
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By placing the two cubes side-by-side in different positions we can form a variety of 2-digit numbers.
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For example, the square number 64 could be formed:
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~~~
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+-------+ +-------+
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/ /| / /|
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+-------+ | +-------+ |
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| | | | | |
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| 6 | + | 4 | +
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| |/ | |/
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+-------+ +-------+
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~~~
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In fact, by carefully choosing the digits on both cubes it is possible to display
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all of the square numbers below one-hundred: `01`, `04`, `09`, `16`, `25`, `36`, `49`, `64`, and `81`.
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For example, one way this can be achieved is by placing `{0, 5, 6, 7, 8, 9}` on one cube
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and `{1, 2, 3, 4, 8, 9}` on the other cube.
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However, for this problem we shall allow the `6` or `9` to be turned upside-down
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so that an arrangement like `{0, 5, 6, 7, 8, 9}` and `{1, 2, 3, 4, 6, 7}` allows
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for all nine square numbers to be displayed; otherwise it would be impossible to obtain `09`.
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In determining a distinct arrangement we are interested in the digits on each cube, not the order.
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- `{1, 2, 3, 4, 5, 6}` is equivalent to `{3, 6, 4, 1, 2, 5}`
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- `{1, 2, 3, 4, 5, 6}` is distinct from `{1, 2, 3, 4, 5, 9}`
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But because we are allowing `6` and `9` to be reversed, the two distinct sets in
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the last example both represent the extended set `{1, 2, 3, 4, 5, 6, 9}` for the
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purpose of forming 2-digit numbers.
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How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed? |