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