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Rachel's Problem

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!


Stage: 4 Challenge Level: Challenge Level:1
James of Hethersett High School, Norfolk came closest to the answer by listing the arrangements and first noting that you can only have one, three, five or seven s's. This is because there are altogether seven letters and the other letters come in pairs. There are altogether 217 words in the Snowman language.

Words with one s
These words have 3 'doubles' which we denote by X, Y and Z, making words in 4 different ways, namely sXYZ, XsYZ, XYsZ or XYZs.
Each of the 'doubles can be any of the 3 choices - no, wm or an - giving 3x3x3=27 choices of XYZ.
Hence there are 4x27=108 words containing one s.

Words with 3 s's
These words have 2 'doubles' X and Y making words in 10 different ways, namely sssXY, ssXsY, ssXYs, sXssY, sXsYs, sXYss, XsssY, XssYs, XsYss, XYsss.
Each of the 'doubles can be any of the 3 choices - no, wm or an - giving 3x3=9 choices of XY.
Hence there are 10x9=90 words containing three s's.

Words with 5 s's
These words have 1 'double', denoted by X, making words in 6 different ways, namely Xsssss, sXssss , ssXsss, sssXss, ssssXs, sssssX

The double can be any of the 3 choices.
Hence there are 6x3=18 words containing five s's.

Word with seven s's
There is one word with seven s's.

For an alternative method, you might like to try to do this by using the fact that words with n letters are either formed by putting an s before words with (n-1) letters or by putting any one of the three 'doubles' before words with (n-2) letters. Using Wn for the number of words with n letters this gives a formula: Wn = Wn-1 + 3Wn-2 where W1= 1 and W2=4.

Using this formula you can find W3, W4, W5, W6 and W7.