This challenge is similar to the following number puzzle, which I found on math.stackexchange, here:

`Given that all the underscores receive one-digit numbers, can you fill this page out so it is true?`

```
+----------------------------------------------+
```

| The number 0 appears _ time(s) on this page. |

| The number 1 appears _ time(s) on this page. |

| The number 2 appears _ time(s) on this page. |

| The number 3 appears _ time(s) on this page. |

| The number 4 appears _ time(s) on this page. |

| The number 5 appears _ time(s) on this page. |

| The number 6 appears _ time(s) on this page. |

| The number 7 appears _ time(s) on this page. |

| The number 8 appears _ time(s) on this page. |

| The number 9 appears _ time(s) on this page. |

+----------------------------------------------+

It's an interesting twist to the number tricks above, mostly because every number appears at least once.

There are a number of interesting explanations on how to get the solution at the question page, but the most intuitive—indeed, the one by which I solved it myself—is detailed below.

(You may wish to pause here in case you want to try it yourself!)

As all the numbers to fill in are restricted to being one digit positive integers, `0`

cannot appear more than once, and all but one of the high numbers (`5`

to `9`

definitely, debatably also `3`

and `4`

) will also appear once, where the odd one out can only appear twice.

`2`

, therefore, will be appear at least twice, once for the "`The number 2 appears`

" part and once in the underscore of the large number. But filling in `2`

with a 2 will mean that you have actually 3 `2`

s.

This can be resolved by giving `3`

a 2 and `2`

a 3; so long as no other numbers besides `2`

, `3`

, and the large one have 2s or 3s, we're safe.

As the number of `1`

s will therefore be one for zero, and one for every number greater than `3`

(except one), we get 1_{(0)} + 1_{(sentence)} + 6_{(>3)} - 1_{(the non-1)} = 7 `1`

s. Thus the `7`

appears twice and the `1`

appears seven times.

The solution is then:

```
+----------------------------------------------+
```

| The number 0 appears __1__ time(s) on this page. |

| The number 1 appears __7__ time(s) on this page. |

| The number 2 appears __3__ time(s) on this page. |

| The number 3 appears __2__ time(s) on this page. |

| The number 4 appears __1__ time(s) on this page. |

| The number 5 appears __1__ time(s) on this page. |

| The number 6 appears __1__ time(s) on this page. |

| The number 7 appears __2__ time(s) on this page. |

| The number 8 appears __1__ time(s) on this page. |

| The number 9 appears __1__ time(s) on this page. |

+----------------------------------------------+

Self-referential puzzles like these are always fun to hand to more casual puzzle-solvers, since they typically have plenty of false starts and get blindsided by the one minute factor they forget about and they end up kicking themselves at the end when they figure it out. Moments of epiphanic realization like those are truly beautiful.