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Logic Puzzles
“Uncle Bob”

Each month in 2020 I have featured a different type of logic puzzle. This month it’s…

Logic: Self-Referential Puzzles

Got your thinking cap on? One characteristic of Self-Referential (S-R) puzzles is that the clue takes an active role in the solution. For example, fill in the blank in the following self-counting sentence with an English word.

“In this sentence the count of e’s is _________.”

You see that the word you fill in may alter the count. Seven works to make a true statement.

Try another:

“In this sentence the number of occurrences of e is _________.”

My First One – another Self-Counting Sentence

What follows is the first S-R puzzle that I ever encountered. It has two solutions. Note that every time you fill in a blank the situation changes.

Fill in the blanks in the following sentence with the correct decimal numerals:

“In this sentence, the number of occurrences of 0 is ____, the number of 1s is ____,

the number of 2s is ____, of 3s is ____, of 4s is ____, of 5s is ____, of 6s is ____, of 7s is ____,

of 8s is ____, and the number of 9s is ____.”

hints and solutions


Angels and Devils S-R Puzzle

I find that Self-Referential puzzles can take many forms. The truther / liar puzzles we sampled back in April often involve self-reference. Here’s one inspired by logician Raymond Smullyan.

The island of Purgatus is inhabited exclusively by Angels and Devils. Angels always tell the truth and Devils always lie. A group of ten islanders were interviewed and made the following claims.

Person 1 said, “Exactly one of us is a liar.”
Person 2 said, “Exactly two of us are liars.”
Person 3 said, “Exactly three of us are liars.”
Person 4 said, “Exactly four of us are liars.”
Person 5 said, “Exactly five of us are liars.”
Person 6 said, “Exactly six of us are liars.”
Person 7 said, “Exactly seven of us are liars.”
Person 8 said, “Exactly eight of us are liars.”
Person 9 said, “Exactly nine of us are liars.”
Person 10 said, “Exactly ten of us are liars.”

Which persons are Angels and which Devils?

hints and solutions


S-R Cross Number Puzzle

Unlike crosswords, cross number puzzles have clues that suggest numerical answers to fill a grid one digit per grid square. They are often assigned in grade school for arithmetic practice. Uncle Bob would not take up your time and mental energy with one of those, no, no, no, but … the topic is Self-Referential puzzles, n’est pas?

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1. 20D x 6
4. 10D x 3
7. 10D – 11A
8. 18A x 4
9. 10 + 1
11. 15A / 3
12. 2 x 3D + 320
14. 9A x 2
15. 2D + 40
16. 8A + 2
18. 11A x 5
21. 8A – 15A – 11A
22. 8A – 14D
1. 20D – 11A
2. 6D / 3
3. 4A x 12 – 160
4. 9A x 5
5. 1A x 9
6. 3 x 5 x 7
10. 18A + 52
12. 21A x 3
13. 8A x 9 + 12
14. 10D + 77
17. 1D – 9A
19. (12A / 4A) + 1
20. 4D + 1

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hints and solutions

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