Glossary

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Sequences and PatternsPascal’s Triangle

Reveal All Steps

Below you can see a number pyramid that is created using a simple pattern: it starts with a single “1” at the top, and every following cells is the sum of the two cells directly above. Hover over some of the cells to see how they are calculated, and then fill in the missing ones:

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
1
12
66
495
792
924
792
495
66
12
1

This diagram only showed the first twelve rows, but we could continue forever, adding new rows at the bottom. Notice that the triangle is symmetricright-angledequilateral, which can help you calculate some of the cells.

The triangle is called Pascal’s triangle, named after the French mathematician Blaise Pascal. He was one of the first European mathematicians to investigate its patterns and properties, but it was known to other civilisations many centuries earlier:

In 450BC, the Indian mathematician Pingala called the triangle the “Staircase of Mount Meru”, named after a sacred Hindu mountain.

In Iran, it was known as the “Khayyam triangle” (مثلث خیام), named after the Persian poet and mathematician Omar Khayyám.

In China, the mathematician Jia Xian also discovered the triangle. It was named after his successor, “Yang Hui’s triangle” (杨辉三角).

Pascal’s triangle can be created using a very simple pattern, but it is filled with surprising patterns and properties. That’s why it has fascinated mathematicians across the world, for hundreds of years.

Finding Sequences

In the previous sections you saw countless different mathematical sequences. It turns out that many of them can also be found in Pascal’s triangle:

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
252
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
1
12
66
220
495
792
924
792
495
220
66
12
1
1
13
78
286
715
1287
1716
1716
1287
715
286
78
13
1
1
14
91
364
1001
2002
3003
3432
3003
2002
1001
364
91
14
1
1
15
105
455
1365
3003
5005
6435
6435
5005
3003
1365
455
105
15
1
1
16
120
560
1820
4368
8008
11440
12870
11440
8008
4368
1820
560
120
16
1

The numbers in the first diagonal on either side are all onesincreasingeven.

The numbers in the second diagonal on either side are the integersprimessquare numbers.

The numbers in the third diagonal on either side are the triangle numberssquare numbersFibonacci numbers.

The numbers in the fourth diagonal are the tetrahedral numberscubic numberspowers of 2.

If you add up all the numbers in a row, their sums form another sequence: the powers of twoperfect numbersprime numbers.

In every row that has a prime number in its second cell, all following numbers are multiplesfactorsinverses of that prime.

The diagram above highlights the “shallow” diagonals in different colours. If we add up the numbers in every diagonal, we get the Fibonacci numbersHailstone numbersgeometric sequence.

Of course, each of these patterns has a mathematical reason that explains why it appears. Maybe you can find some of them!

Another question you might ask is how often a number appears in Pascal’s triangle. Clearly there are infinitely many 1s, one 2, and every other number appears at least twiceat least onceexactly twice, in the second diagonal on either side.

Some numbers in the middle of the triangle also appear three or four times. There are even a few that appear six times: you can see both 120 and 3003 four times in the triangle above, and they’ll appear two more times each in rows 120 and 3003.

Since 3003 is a triangle number, it actually appears two more times in the third diagonals of the triangle – that makes eight occurrences in total.

It is unknown if there are any other numbers that appear eight times in the triangle, or if there numbers that appear more than eight times. The American mathematician David Singmaster hypothesised that there is a fixed limed on how often numbers can appear in Pascal’s triangle – but it hasn’t been proven yet.

Divisibility

Some patterns in Pascal’s triangle are not quite as easy to detect. In the diagram below, highlight all the cells that are even:

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1

It looks like the even number in Pascal’s triangle form another, smaller trianglematrixsquare.

Colouring each cell manually takes a long time, but here you can see what happens if you would do this for many more rows. And what about cells divisible by other numbers?

1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
1
8
28
56
70
56
28
8
1
1
9
36
84
126
126
84
36
9
1
1
10
45
120
210
252
210
120
45
10
1
1
11
55
165
330
462
462
330
165
55
11
1
1
12
66
220
495
792
924
792
495
220
66
12
1
1
13
78
286
715
1287
1716
1716
1287
715
286
78
13
1
1
14
91
364
1001
2002
3003
3432
3003
2002
1001
364
91
14
1
1
15
105
455
1365
3003
5005
6435
6435
5005
3003
1365
455
105
15
1
1
16
120
560
1820
4368
8008
11440
12870
11440
8008
4368
1820
560
120
16
1
1
17
136
680
2380
6188
12376
19448
24310
24310
19448
12376
6188
2380
680
136
17
1
1
18
153
816
3060
8568
18564
31824
43758
48620
43758
31824
18564
8568
3060
816
153
18
1
1
19
171
969
3876
11628
27132
50388
75582
92378
92378
75582
50388
27132
11628
3876
969
171
19
1
1
20
190
1140
4845
15504
38760
77520
125970
167960
184756
167960
125970
77520
38760
15504
4845
1140
190
20
1
1
21
210
1330
5985
20349
54264
116280
203490
293930
352716
352716
293930
203490
116280
54264
20349
5985
1330
210
21
1
1
22
231
1540
7315
26334
74613
170544
319770
497420
646646
705432
646646
497420
319770
170544
74613
26334
7315
1540
231
22
1
1
23
253
1771
8855
33649
100947
245157
490314
817190
1144066
1352078
1352078
1144066
817190
490314
245157
100947
33649
8855
1771
253
23
1
1
24
276
2024
10626
42504
134596
346104
735471
1307504
1961256
2496144
2704156
2496144
1961256
1307504
735471
346104
134596
42504
10626
2024
276
24
1

Wow! The coloured cells always appear in trianglessquarespairs (except for a few single cells, which could be seen as triangles of size 1).

If we continue the pattern of cells divisible by 2, we get one that is very similar to the Sierpinski triangle on the right. Shapes like this, which consist of a simple pattern that seems to continue forever while getting smaller and smaller, are called Fractals. You will learn more about them in the future…

Sierpinski Triangle

The Sierpinski Triangle

Binomial Coefficients

There is one more important property of Pascal’s triangle that we need to talk about. To understand it, we will try to solve the same problem with two completely different methods, and then see how they are related.

COMING SOON