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Solve my base three numerical self-referential sentences puzzle!

Jennifer Franklin of Ottawa, Canada was the first to solve this puzzle. Franklin submitted her solution on October 29, 1998. This page was created in March 1997. I'm surprised that a year and a half elapsed before someone found a correct answer!

What are self-referential sentences?

Self-referential sentences are sentences that comment on their own characteristics, structure, and function. For example, this sentence is self-referential because it asserts its self-referential nature.

My puzzle is a system of four numerical self referential sentences; each sentence in the set refers to its digital or numerical contents and/or the digital or numerical contents of other sentences in the set. This particular system of sentences is different than typical systems of self-referential sentences because all the numbers are in base three.

What are base three numbers?

Base three numbers are the same as familiar decimal numbers except that they are written with only three different digits instead of ten. Each digit in a number represents that multiple of the number's base to a power determined by the digit's position in relation to the number's decimal point. An entire number is equal to the sum of these digital multiples of successive powers of the base. For example, the base ten number 5372 = 5 × 103 + 3 × 102 + 7 × 101 + 2 × 100. It's easy to convert numbers of any base to base ten. You simply multiply each digit by the appropriate power of the base and add up the results. For example, the base three number 2201 = 2 × 33 + 2 × 32 + 0 × 31 + 1 × 30 = 73. It is also fairly simple to obtain a base three number from a base ten number by dividing the number by three, generating a digit from the remainder, and repeating the process with the integer of the quotient to obtain successive digits. For example, we could reverse the example shown above (R stands for remainder):

73 ÷ 3 = 24 R 1
24 ÷ 3 = 8 R 0
8 ÷ 3 = 2 R 2
2 ÷ 3 = 0 R 2

The base ten number 73 is thus converted back to the base three number 2201.

How to solve this puzzle

The object of this puzzle is to fill in the fields in each sentence with positive base three numbers so that all of the sentences are true. Remember that all the numbers already in the sentences are also in base three. The red numbers serve simply to number the sentences in base three from one to four and should not be considered part of the sentences. When a sentence refers to the number of 0s, 1s, or 2s in a sentence or a set of sentences, it means the number of occurrences of that particular digit, not the number of occurrences of that particular number. So in its initial state sentence 1. has five (written "12") 1s, not two.

This puzzle may have more than one solution. See if you can find a solution other than the one on the solutions page.


Here's the puzzle:

1. Sentences 1, 2, 10, and 11 together have a total of 0s, 1s, and 2s.

2. Sentences 2, 10, and 11 together have a total of 0s, 1s, and 2s.

10. Sentence has 20 more 1s than this sentence, which has only 1s.

11. The sum of all the numbers (e.g. 2 + 11 + 10 = 100) in sentences 1, 2, 10, and 11, minus the last number in this sentence, is .


Check out David Dewey's home page!

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