Counting on Letters

The basis of this investigation comes from discussion with some teachers in Hungary, England and Denmark, who met through our Euromaths Project, and was adapted by Merilyn Buchanan.

The letters of the word ABACUS have been arranged in the shape of a triangle.

A
B B
A A A
C C C C
U U U U U
S S S S S S

What do you notice about the pattern?

How many different ways can you find to read the word ABACUS from this triangular pattern?

If the shape of the triangle were changed to the one below, how does the pattern of the letters change?

A
B B B
A A A A A
C C C C C C C
U U U U U U U U U
S S S S S S S S S S S

Are there more or fewer ways to read the word ABACUS in this triangle?

Try arranging the letters ABACUS into a different shape triangle.

Can you see a pattern in the letters of the new triangle that is similar to either of the triangles above?

Record the number of ways to read the word ABACUS in your triangle. Are there more or fewer ways to read the word ABACUS in your triangle than in the other triangles?

You can find information about another triangle, Pascal's Triangle in this article.Can you find any similarities between these triangles and the patterns in Pascal's triangle?

The letters could also be arranged in a rectangle like this:

A	B	A	C
B	A	C	U
A	C	U	S

If you begin in the top left hand corner and move only to the right or immediately below the letter, how many ways are there to read the word ABACUS?

Arrange the letters of ABACUS into other size rectangles.
How many ways can you find to read the word ABACUS? Is there any pattern between the number of ways you can find the word and the size or shape of the rectangle?

Try the same problems with the word PUMPKIN.