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Difficulty: Fun

Triangular numbers are numbers that can be formed into a triangle.

For example, 3 is a triangular number with 2 dots on the bottom row and 1 dot on top row forming an equilateral triangle of 3 dots. 6 is the next triangular number with 3 dots on the bottom, 2 dots in the middle, and 1 dot on top forming a triangle of 6 total dots. If we add 4 dots underneath 6’s triangle, we make a new, bigger triangle of 10 dots, like so:

If we start with one as our first triangle, each triangle after that adds one more dot to the row beneath. Our second triangle adds 2 dots to the row beneath one dot to make 3.

Given the pattern described above, what is the 10th (T₁₀) triangular number?

Bonus question: 2 consecutive (right after each other) triangular numbers always add up to a square number. For example, T₁ + T₂ = 1 + 3 = 4 (the square of 2). Can you use geometry to show why this happens?

Scroll down or click for a hint, or the answer!

Brainteaser hint

First, use the pattern described above or illustrated in the picture to find out how many dots will be on the bottom row of the T₁₀ triangle. Then, ask how many dots will be on the next row above, and the one above that and so on.

For the bonus question: Remember that a square number of dots can be arranged in a square grid.

Try visualising the triangular numbers as right triangles so that the dots in each triangle look like a set of stairs, like this:

Brainteaser answer

The tenth triangular number, T₁₀, is 55.

The best way to look at these triangles is one layer at a time. Starting at the bottom, each layer has one fewer dot than the last. When you reach the top, it only has only 1 dot.

The size of the base is quite simple – the first triangular number has a base of 1, the second a base of 2 and so on. Extending this pattern, the 10th triangular number will have a base of 10 dots.

The most straightforward way to find T₁₀ is to add up the rows starting with 10. Each subsequent row will have one less dot in it than the previous row. So, adding up the rows looks like this:

T₁₀ = 10 + 9 + 8 +7 + 6 + 5 + 4 + 3 + 2 + 1 = 55

Bonus question: Two consecutive (right after each other) triangular numbers always add up to a square because these 2 triangles literally fit together to make a square!

Remember that a square number is the result of any whole number multiplied by itself. We call it a square number because squares have equal sides and so the area of a square is always length times equal length. A number multiplied by itself. Examples of square numbers include 1, 4, 9, 16, 25, 36, and so on.

To see how 2 consecutive triangular number become a square number, first imagine the triangular numbers as right triangles. This way the dots of each triangle are arranged like a set of stairs:

Next, imagine rotating the smaller triangle 180 degrees and fitting it into the larger triangle:

Using geometry, we can see that 2 consecutive triangular numbers make a square. In other words, they add up to a square number. On the left we see that 1 and 3 fit together to make a 2×2 square where the dots add up to the square number 4 (or 2²). On the right we see that 6 and 10 make a 4×4 square where the dots add up the square number 16 (or 4²).

If you’re not convinced by the geometry, here’s an equation-based explanation. We’ll use the triangles of 8 and 7 as an example, but this method will work for any 2 consecutive triangle numbers.

T₈ = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
T₇ = 7 + 6 + 5 + 4 + 3 + 2 + 1

We can add (+ 0) to the T₇ formula so that both equations have 8 terms, and then re-order the sum from smallest to largest:

T₇ = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7

And then we can add T₈ and T₇, term by term:

T₈ + T₇ = (8 + 0) + (7 + 1) + (6 + 2) … + (1 + 7)
T₈ + T₇ = 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8
T₈ + T₇ = 8 x 8