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Triangular Numbers

If we place coins touching in a row, we can complete rows above with more coins each coin touching two on the row below.

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1 3 6 10

This gives a series of triangular shapes and the number of dots in each shape is a triangular number. So the series of triangular numbers begins:

1, 3, 6, 10, 15, 21, ...

Can you spot the pattern behind this sequence and so provide the next two numbers in the list without drawing the dots and counting them?

The general formula for the nth triangular number is: n (n + 1)

2

So the hundredth triangular number is: 100 101 = 5050

2

The triangular numbers have many interesting number properties. For instance, add two neighbouring triangular numbers - what do you notice? Starting from the beginning and adding pairs we get the series:

1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, ...

which are the square numbers. This is made clearer by putting the two neighbouring triangles together to form a square:

1 + 3 = 22 3 + 6 = 32 6 + 10 = 42

Suppose we use just the triangular numbers to make addition sums whose total is one of the non-triangular numbers. Then we have:

  • 1 is a triangular number!
  • 2 is 1+1, both triangular numbers;
  • 3 is a triangular number!
  • 4 is 1+3, both triangular numbers;
  • 5 is 1+1+3 and all 3 are triangular numbers;
  • 6 is a triangular number!
  • 7 is 1+6, both triangular numbers;
  • 8 is 1+1+6, and all 3 are triangular numbers;
  • 9 is 3+6, both triangular numbers;
  • 10 is a triangular number!

So we can make a game: Pick a whole number. Can you find up to 3 triangular numbers which add to your chosen value? From the table above, it seems we can do so for all the numbers up to 10 anyway. In fact, it can always be done:

Every number is a sum of at most 3 triangular numbers

Can you...

find how to do this for all numbers up to 20? up to 30? up to 100?

Pick any triangular number. Can you use 8 copies of it as 8 jigsaw puzzle pieces to make a square (number)?

Pick any triangular number. Can you use 3 copies of it together with one of the next larger triangular number to make a third triangular number?


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