What patterns can you find in Pascal’s triangle?
February 6th, 2008 | by Sol |Pascal’s triangle is a very simple thing to construct, yet it has a tremendous amount of depth to it. Also, Pascal’s triangle has a
huge number of interesting patterns in it. If you’re not familiar with Pascal’s triangle then read this background article at Wikipedia and then try your hand at finding some of the following patterns in the numbers, or in sums of numbers, or in some other relationship among numbers. The patterns might appear in a line going across the triangle, along one of the diagonals or elsewhere.
There are a couple of resources listed at the end of this article that give a number of the answers but try to find them yourself before checking the references.
1. Consecutive natural numbers: 1, 2, 3, 4, … Where can you find this sequence in the triangle? It appears twice as do many of the patterns.
2. Triangular numbers: Those are sums of consecutive integers starting with 1. So, 1 is a triangular number, so is 1+2, 1+2+3, 1+2+3+4, and so on. Where can you find these twice?
3. Can you see a pattern to the sum of the numbers in any row of the triangle?
4. Can you find a pattern in the alternating sum of the numbers in any row of the triangle? For example, 1-4+6-4+1, for the 5th row of the triangle?
5. Can you find powers of 11 in Pascal’s Triangle?
6. Fibonacci numbers. They form a sequence, starting with 1, 1 and each number is the sum of the two before it. So, the first few terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55. This is a tricky pattern to find in the triangle. You’ll find numbers that add up to terms of the sequence but they may not be easy to find.
7. Squares. Can you find a pattern of pairs of consecutive entries in the triangle that add up to consecutive perfect squares?
8. If the second element in a row of Pascal’s triangle (the one after the 1) is a prime number what can you tell about the other entries in that row, excluding the 1’s?
Resources:
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5 Responses to “What patterns can you find in Pascal’s triangle?”
By Jonathan on Feb 9, 2008 | Reply
I had students readier for Fibonacci this year than ever before. Early in the fall we counted the number of ways a rabbit could hop up a flight of 12 steps, one or two at a time. Instead of the standard (solve same problem for a small staircase, build up, notice pattern), we had a group divide the problem into 6 doubles, 5 doubles 2 singles, 4d 4s, 3d 6s, 2d 8s, 1d 10s, 12s. They took the sum C(6,6), C(7,5), C(8,4), C(9,3), C(10,2), C(11,1), C(12,0)
So, two months later, when we looked at Pascal’s triangle, how easy were the corresponding entries to spot?
Jonathan
By Sol on Feb 10, 2008 | Reply
Jonathan,
That’s a great story. I really enjoy hearing how you inspire your students. And, yes, I can see how after all of the work with Fibonacci earlier that spotting it in Pascal’s triangle was a breeze!
By Rebecca on Feb 15, 2008 | Reply
A really fun website for “integer-sequence-philes” is The On-Line Encyclopedia of Integer Sequences. Any sequence you find in Pascal’s triangle probably has a name and you can find it (along with a lot of other information such as a generating formula and papers citing that sequence) there.
By Sol on Feb 16, 2008 | Reply
Rebecca, I agree. That’s a really great resource. I hadn’t made the connection between Pascal’s triangle and the Integer Sequences site. Thanks for the suggestion. I’ll mention it next time I write about something related to sequences.
By Jonathan on Feb 17, 2008 | Reply
Sol, I can’t find your e-mail, so I am sort of cheating. I’ll be hosting the Carnival of Mathematics Friday, and was hoping you’d submit something. E-mail me, or use the tool from my site.
Thanks!