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

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 Brain Teasers   |   Math
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Description

"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 the blanks with the numerical digits (1, 2, etc.) so that the sentence is true. Can you find the 2 solutions? Sources And Further Reading ↓↓↓

The puzzle is due to the logician Raphael M. Robinson and it was popularized by Douglas Hofstadter in his book "Metamagical Themas."

My blog post for this video
http://wp.me/p6aMk-4X7

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

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My Books

"The Joy of Game Theory" shows how you can use math to out-think your competition. (rated 3.9/5 stars on 24 reviews) https://www.amazon.com/gp/product/1500497444

"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 5/5 stars on 2 reviews) https://www.amazon.com/gp/product/1523231467/

"Math Puzzles Volume 1" features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.5/5 stars on 11 reviews. https://www.amazon.com/gp/product/1517421624/

"Math Puzzles Volume 2" is a sequel book with more great problems. (rated 5/5 stars on 3 reviews) https://www.amazon.com/gp/product/1517531624/

"Math Puzzles Volume 3" is the third in the series. (rated 5/5 stars on 3 reviews) https://www.amazon.com/gp/product/1517596351/

"40 Paradoxes in Logic, Probability, and Game Theory" contains thought-provoking and counter-intuitive results. (rated 4.8/5 stars on 9 reviews) https://www.amazon.com/gp/product/1517319307/

"The Best Mental Math Tricks" teaches how you can look like a math genius by solving problems in your head (rated 4.7/5 stars on 3 reviews) https://www.amazon.com/gp/product/150779651X/

"Multiply Numbers By Drawing Lines" This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 5/5 stars on 2 reviews) https://www.amazon.com/gp/product/1500866148/

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