In this sentence, the number of occurrences

of the digit 0 is __,

of the digit 1 is __,

of the digit 2 is __,

of the digit 3 is __,

of the digit 4 is __,

of the digit 5 is __,

of the digit 6 is __,

of the digit 7 is __,

of the digit 8 is __, and

of the digit 9 is __.

The challenge is to fill in the blanks with numerical digits (1, 2, etc.) so that the sentence is true.

There are 2 solutions. Can you find them?

Watch the video for a solution.

**Can You Solve The Mind-Bending Self-Counting Sentence Riddle? **

Or keep reading.

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**Answer To The Mind-Bending Self-Counting Sentence Riddle**

The answers can be found by experimentation. To start, each of the digits appears at least once, so fill in 1 for every line (except the digit 1 which is left blank at the moment).

TEST CASE 1

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is __,

of the digit 2 is 1,

of the digit 3 is 1,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 1.

How many times does the digit 1 occur? The sentence already contains 10 occurrences of the digit 1, so let us try that value.

TEST CASE 2

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 10,

of the digit 2 is 1,

of the digit 3 is 1,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 1.

This sentence is false, however, because the digit 1 appears 11 times in total (including the “1” in the number “10.”). We can try to adjust the sentence to 11 occurrences of the digit 1.

TEST CASE 3

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 11,

of the digit 2 is 1,

of the digit 3 is 1,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 1.

But now this sentence is also false because it contains 12 occurrences of the digit 1 (the number “11” adds another “1” to the count). If we increase the count to 12, then the occurrences of the digit 2 would increase to 2, thereby reducing one of the occurrences of the digit 1. So let’s try keeping the occurrences of the digit 1 as 11, and increasing the occurrences of the digit 2 to be 2.

TEST CASE 4 = Solution One

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 11,

of the digit 2 is 2,

of the digit 3 is 1,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 1.

This sentence is true and is the first solution to the puzzle. The digit 1 occurs 11 times, the digit 2 occurs 2 times, and every other digit occurs 1 time.

But there’s another solution this problem too! Let us adjust the above solution by considering if the digit 1 only has 9 occurrences to avoid having a double-digit number.

TEST CASE A

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 9,

of the digit 2 is 2,

of the digit 3 is 1,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 1.

The sentence if false because there are 2 occurrences of the digit 9. If the digit 9 occurs 2 times, then that means the digit 2 has to occur 3 times total. And that also means the digit 3 occurs 2 times.

TEST CASE B

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 9,

of the digit 2 is 3,

of the digit 3 is 2,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 1,

of the digit 8 is 1, and

of the digit 9 is 2.

This sentence is false; now there are only 7 occurrences of the digit 1. To fix the sentence, we can swap the occurrences of the digits 7 and 9. That is, write that the digit 7 occurs 2 times, and write that the digit 9 occurs 1 time.

TEST CASE C = Solution Two

In this sentence, the number of occurrences

of the digit 0 is 1,

of the digit 1 is 7,

of the digit 2 is 3,

of the digit 3 is 2,

of the digit 4 is 1,

of the digit 5 is 1,

of the digit 6 is 1,

of the digit 7 is 2,

of the digit 8 is 1, and

of the digit 9 is 1.

This sentence is true and it is the second solution. It is in fact the unique solution if all the digits occur less than 10 times.

And there you have it: two true sentences that count themselves!

**Sources and further reading**

Douglas Hofstadter *Metamagical Themas* on Google Books

https://books.google.com/books?id=o8jzWF7rD6oC&lpg=PA390&ots=jRCg7rLufs&dq=The%20number%20of%200s%20in%20this%20sentence%20Metamagical%20Themas&pg=PA390#v=onepage&q&f=false

Math Central December 2002 Problem. Proof there are only 2 solutions

http://mathcentral.uregina.ca/mp/archives/previous2002/dec02sol.html

Self-descriptive sentences (and two solutions described)

http://www.cut-the-knot.org/ctk/SelfDescriptive.shtml

Unique solution if all occurrences less than 10

http://math.stackexchange.com/questions/19061/puzzle-digit-x-appears-y-times-on-this-piece-of-paper?rq=1

Sketch of proof that there are only 2 solutions

https://www.reddit.com/r/math/comments/2oh9s9/heres_a_maths_puzzle_my_friend_posted_on_facebook/cmn8f26/

Solving by iterative process

http://web.archive.org/web/20120428023510/http://www.lboro.ac.uk/departments/ma/gallery/selfref/index.html

Variations for self-counting sentences

http://lkozma.net/blog/self-counting-sentences/

http://lkozma.net/blog/self-counting-sentences-ii/

Autogram (self-counting with letters)

https://en.wikipedia.org/wiki/Autogram

Variation with letters (autogram)

http://www.braingle.com/news/hallfame.php?path=language/english/sentences/self.ref/self.ref.letters.p&sol=1

More here:

Can You Solve The Mind-Bending Self-Counting Sentence Riddle? Sunday Puzzle