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A large triangle of dots on a night sky, odd numbers in gold and even in blue so that triangles appear inside it, beside the cover of Escape the Crystal Mine.

Field Notes

Pascal's Triangle: Patterns Hiding in a Number Pyramid

4 min read
  • Number patterns
  • Pyramids
  • Maths for fun

Build a number pyramid where every brick is the sum of the two bricks above it, with 1s down both edges. That is Pascal's triangle. It looks plain, yet it hides a surprising number of patterns. Here are five, each one drawn so you can check it yourself.

Where it comes from

It is named after Blaise Pascal, who wrote a treatise on it in the 1650s, but it is much older. Commentators on the Sanskrit work Chandas Shastra drew it around the 10th century (Halayudha, around the year 1000), the Persian poet and mathematician Omar Khayyam is linked with it (in Iran it is called the Khayyam triangle), and in China it is called Yang Hui's triangle after a 13th-century mathematician (historical note). Many people found it; Pascal's name stuck in the West.

The rule

A six-row triangle of numbers. The two 3s in row 3 are green and the 6 beneath them is orange, showing 3 plus 3 equals 6.
Each brick is the sum of the two bricks above it. The next row is 1, 5, 10, 10, 5, 1.

If you have met the first four rows before, it was probably in our post The Top-Brick Shortcut, where 1, 3, 3, 1 tells you how many times each bottom brick counts. This post looks at the whole triangle.

Pattern 1: each row adds up to a power of 2

A seven-row triangle with each row's total written beside it: 1, 2, 4, 8, 16, 32 and 64.
Row totals double every time.

Add row 4: 1 + 4 + 6 + 4 + 1 = 16. The totals go 1, 2, 4, 8, 16, 32, 64. The reason: every brick is passed down into two bricks of the next row, so the next row's total is exactly double.

Pattern 2: counting numbers and triangular numbers

An eight-row triangle. The second diagonal is orange and reads 1 to 7. The third diagonal is blue and reads 1, 3, 6, 10, 15, 21.
Orange: the counting numbers. Blue: the triangular numbers.

Walk down the second diagonal and you meet 1, 2, 3, 4, 5, 6, 7. The next diagonal gives 1, 3, 6, 10, 15, 21, the triangular numbers: 1, then 1 + 2, then 1 + 2 + 3, and so on. They are the number of dots in a triangle of dots, and the triangle builds them for you, because each blue brick is the blue brick above it plus the orange brick next to that.

Pattern 3: slanted diagonals give Fibonacci numbers

An eight-row triangle with bricks coloured along slanted lines. The sums of each line are 1, 1, 2, 3, 5, 8, 13 and 21.
Add the bricks on each slanted line.

Now cut the triangle on a shallow slant. Take 1 + 5 + 6 + 1 = 13, or 1 + 6 + 10 + 4 = 21. The slanted sums run 1, 1, 2, 3, 5, 8, 13, 21: the Fibonacci numbers, where each number is the two before it added together.

Pattern 4: colour the odd numbers

On the left, eight rows with odd numbers in green and even numbers in white. On the right, the first 32 rows showing only the odd bricks, forming triangles inside triangles.
Odd numbers only, first 8 and then 32 rows.

Shade every odd brick and leave the even ones empty. Triangles appear, with smaller triangles of empty bricks inside bigger ones. Carry on for more rows and you approach a famous shape, the Sierpinski triangle (Chalkdust article). You only ever need to know whether a sum of two numbers is odd or even to draw it.

Pattern 5: it counts routes

A five-row triangle joined by lines. The orange 6 at the bottom has six routes listed: LLRR, LRLR, LRRL, RLLR, RLRL and RRLL.
Start at the top and move down-left (L) or down-right (R).

Start at the top and, at each step, go down-left or down-right. The number in a brick is how many different routes reach it. Four steps ending in the orange 6 can be written with two Ls and two Rs in six different orders. Flipping a coin four times works the same way: 6 of the 16 possible results have exactly two heads. That is why this triangle turns up in chances and choices.

Try it yourself

  • Write out 10 rows. Check each row total is a power of 2.
  • Find the Fibonacci sums on the slanted lines.
  • Shade the odd numbers with a pencil and see the triangles appear.
  • Pick any brick and count its routes from the top by hand.

Where a puzzle adventure fits

Number pyramids with sums are one of the puzzle kinds in Escape the Pyramid, a puzzle adventure gamebook, so you do not read it in order. Pascal's triangle itself is not in the book: it is the same add-the-two-bricks idea, grown into a pattern. Level 1 (Easy) and Level 3 (Hard) each have 45 locked chambers and 6 treasure rooms, with hints and full answers in the back.

A chamber page of Escape the Crystal Mine: a drawn mine tunnel with a door, the Crystal Clusters rule, a partly filled number pyramid and a Next Door box.
A chamber page as printed, at low resolution. Nothing is solved.

Books mentioned

Escape the Pyramid, Level 1 (Easy) cover

Escape the Pyramid, Level 1 (Easy)

A puzzle adventure gamebook with 45 locked chambers and 6 treasure rooms, hints and full answers in the back.

View on Amazon
Escape the Pyramid, Level 3 (Hard) cover

Escape the Pyramid, Level 3 (Hard)

A puzzle adventure gamebook of 45 harder chambers and 6 treasure rooms, with hints and full answers in the back.

View on Amazon
Escape the Crystal Mine cover

Escape the Crystal Mine

Go down nine glittering levels through 45 locked chambers. A puzzle adventure gamebook.

View on Amazon

As an Amazon Associate we earn from qualifying purchases. Cover images are the ones we have on file; the book you receive may show a slightly different cover.

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