In an LCD display some numbers, when viewed upside-down, are images of other numbers. For example, 1995 becomes 5661. The fifth number that can be read upside down is 8, and the 15th is 21, which is 12 when viewed upside-down. What is the millionth number that is meaningful upside-down?

The number in question are those that contain only 0, 1, 2, 5, 6, 8 and 9--seven distinct digits.

If we use base-7 numbers as a representation of the ordinal positions of the chosen decimal numbers, we can merely take the base-7 version of the ordinal number and translate 3->5, 4->6, 5->8 and 6->9:

ordinal  base-7   translation
1          1           1
2          2           2
3          3           5
4          4           6
5          5           8
6          6           9
7         10          10
8         11          11
9         12          12
10        13          15
11        14          16
12        15          18
13        16          19
14        20          20
15        21          21
16        22          22

This works because the numerical sequence of the number sets correspond to one another, as we just kept the digits in numeric order.

1,000,000 in base-7 is 11333311, so the translation then becomes 11555511, as the millionth number that's meaningful upside-down. It also happens to have the same value upside down.

Posted by Charlie on 2013-07-15 12:52:44