Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's
there is one digit, between the two 2's there are two digits, and
between the two 3's there are three digits.
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 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?
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
Try arranging the letters ABACUS into a different shape
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?
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.