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Problem of the Week
Problem A and Solution
Numbered Triangle


Diana has outlined the following triangle formed of boxes:

A tower of identical boxes arranged into 9 rows. 9 boxes lying side by side form the bottom row. Every row above the bottom row contains one less box than the row below it, ending with 1 box in the top row. Every box in the tower touches exactly two boxes in the row below it, except for the boxes in the bottom row.

Diana fills in the top box of the triangle with a 1. The first box and last box of the remaining rows of the triangle are filled with a 1. The rest of the boxes in each row are filled with the sum of the numbers in the two boxes in the row above that touch the top side of the box Diana is filling. For example, the middle box of the third row is filled with the sum of the numbers in the two boxes in the second row.

What does the triangle look like after Diana has filled in all of the numbers according to the rules described above?

Can you find any patterns in the finished triangle? Share these patterns with others in your class.

Not printing this page? You can fill in the boxes on our interactive worksheet.


Here is the completed triangle:

The top three boxes have the number 1. The numbers in the boxes in the third row from the top are 1, 2, 1, in that order. The fourth row is 1, 3, 3, 1. The fifth row is 1, 4, 6, 4, 1. The sixth row is 1, 5 ,10 ,10, 5, 1. The seventh row is 1, 6, 15, 20, 15, 6 ,1. The eighth row is 1, 7, 21, 35, 35, 21, 7, 1, and the ninth row is 1, 8, 28, 56, 70, 56, 28, 8, 1.

Teacher’s Notes

This problem is an exploration of Pascal’s Triangle which is named for French mathematician Blaise Pascal (1623 - 1662). There are lots of patterns to be found in this structure. It is easier to describe many of the patterns in terms of rows of the triangle, in particular we refer to the top row of the triangle as Row 0. Here are a few of the patterns:

The list above is not comprehensive, there are many other patterns in Pascal’s Triangle. Hopefully your students were able to find some others!