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! |

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.

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* × 10^{3} + *3* × 10^{2} + *7* × 10^{1} + *2* × 10^{0}.
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* × 3^{3} + *2* × 3^{2} + *0* × 3^{1} + *1* × 3^{0} = *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.

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
*0*s,
*1*s, or
*2*s 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") *1*s, 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.