Pattern detectives: predict, test, explain

In the first two layers you learned to spot rules and use them. Now you become a pattern detective. Detectives do not just guess; they predict, test, check and explain.

This layer is full of investigations. Each one starts with a question like “how many sticks for 100 squares?”, “does this trick always work?” or “which grows faster?”. You will make a prediction first (and write it down, so you cannot pretend afterwards!), then test it with the labs, tables and a pencil, and finally try to explain why the answer comes out the way it does.

Some of your predictions will be wrong. That is the point. A wrong prediction you can explain teaches you more than a lucky right one.

Investigation: two rows of squares. Build a rectangle of matchstick squares 2 rows high and n squares long. Count the sticks for n = 1, 2, 3, 4, 5.

Squares long (n)	1	2	3	4	5
Sticks	7	12	17	22	27

The differences are all 5, so the rule is 5 × n + something. For n = 1 we need 7, so the rule is 5 × n + 2.

Why 5? Each new column adds 3 horizontal sticks (top, middle, bottom) and 2 vertical sticks: 5. Why + 2? The 2 vertical sticks on the far left edge.

Try 3 rows high yourself before reading on. (You should get 10, 17, 24, 31, …, which is 7 × n + 3: 4 horizontal and 3 vertical sticks per column, plus 3 on the left edge.)

Investigation: how many dots in the 100th triangle? Adding 1 + 2 + 3 + … + 100 by hand is slow. Here is a trick you can test with counters.

Make a triangle of dots with rows 1, 2, 3, 4 (10 dots). Make a second, identical triangle and turn it upside down. Push the two together: they make a rectangle 4 dots wide and 5 dots tall, which has 4 × 5 = 20 dots. One triangle is half of that: 10. ✓

Triangle n	Rectangle	Rectangle dots	Triangle dots (half)
3	3 × 4	12	6
4	4 × 5	20	10
5	5 × 6	30	15
10	10 × 11	110	55
100	100 × 101	10,100	5,050

So the nth triangular number is n × (n + 1) ÷ 2.

Why do odd numbers build squares? Picture a 1 by 1 square of one tile. To make it into a 2 by 2 square, wrap an L-shape of 3 tiles around two sides. To grow that into a 3 by 3 square, wrap an L of 5 tiles. Then an L of 7 tiles makes a 4 by 4 square.

Step	L-shape added	Square made	Total tiles
1	1 (just the corner)	1 × 1	1
2	3	2 × 2	1 + 3 = 4
3	5	3 × 3	4 + 5 = 9
4	7	4 × 4	9 + 7 = 16
5	9	5 × 5	16 + 9 = 25

Each L has one tile on the corner and two equal arms. To go from an (n − 1) square to an n square, each arm has n − 1 tiles, so the L has (n − 1) + (n − 1) + 1 = 2 × n − 1 tiles: exactly the nth odd number. These L-shapes are the growing L-shapes in the pattern machine. The Greeks called them gnomons, the name for the L-shaped pointer on a sundial.

A famous story is told about Carl Friedrich Gauss, who became one of the greatest mathematicians in history. When he was about nine, his teacher asked the class to add up all the numbers from 1 to 100, hoping for some peace and quiet. Gauss wrote the answer almost immediately: 5,050.

Treat that as a story, not history. It is an anecdote — historians class it as apocryphal — that has grown in the retelling, and the pairing trick itself was written down centuries before Gauss was born. What is real is the idea, so here it is.

Pair the numbers from the two ends:

• 1 + 100 = 101
• 2 + 99 = 101
• 3 + 98 = 101
• …
• 50 + 51 = 101

There are 50 pairs, each adding to 101, so the total is 50 × 101 = 5,050.

This is the same as the two-staircase rectangle from the dot chapter: 100 × 101 ÷ 2 = 5,050.

Investigation: the 2 × 2 box. On a hundred square, draw a 2 by 2 box around four numbers, say 23, 24, 33, 34. Multiply the numbers on each diagonal:

• 23 × 34 = 782
• 24 × 33 = 792

The difference is 10. Try another box: 56, 57, 66, 67 gives 56 × 67 = 3752 and 57 × 66 = 3762: again a difference of 10.

Now try it on a calendar, where rows have 7 days. The box 8, 9, 15, 16 gives 8 × 16 = 128 and 9 × 15 = 135: difference 7.

Conjecture: the difference is always the row length: 10 on the hundred square, 7 on a calendar. In the Deepen layer you can prove it with a little algebra: if the top-left number is a and the rows have length r, the box is a, a + 1, a + r, a + r + 1, and (a + 1)(a + r) − a(a + r + 1) = r.

Investigation: the 3 × 3 box. In Discover you met the calendar trick: the nine dates in any 3 by 3 box add up to 9 × the middle date. Why?

Look at the box around a middle date m on a calendar:

m − 8	m − 7	m − 6
m − 1	m	m + 1
m + 6	m + 7	m + 8

Every number above or to the left of the middle has a partner below or to the right that is the same amount bigger: m − 8 pairs with m + 8, m − 7 with m + 7, m − 6 with m + 6, m − 1 with m + 1. The pluses and minuses cancel, leaving nine m's: 9 × m.

The same works on the hundred square (with 11, 10, 9 and 1 instead of 8, 7, 6 and 1), and for any 3 × 3 box on any grid.

Investigation: multiplying a two-digit number by 11. Try a few:

Number	× 11	Notice
23	253	2 _ 3 with 2 + 3 = 5 in the middle
45	495	4 _ 5 with 4 + 5 = 9 in the middle
61	671	6 _ 1 with 6 + 1 = 7 in the middle
78	858	7 + 8 = 15: write 5, carry 1 into the 7
99	1,089	9 + 9 = 18: write 8, carry 1 into the 9

Conjecture: to multiply ab by 11, write a, then a + b, then b. Test: it works whenever a + b is 9 or less. When a + b is 10 or more, you must carry. So the neat version is only sometimes true, but the version with carrying is always true, because ab × 11 = ab × 10 + ab.

Investigation: who saves more? Arjun saves ₹500 every week. Bela saves ₹1 in week 1, then doubles what she saves each week: ₹1, ₹2, ₹4, ₹8, …

Week	Arjun that week	Bela that week	Arjun total	Bela total
1	₹500	₹1	₹500	₹1
5	₹500	₹16	₹2,500	₹31
9	₹500	₹256	₹4,500	₹511
10	₹500	₹512	₹5,000	₹1,023
12	₹500	₹2,048	₹6,000	₹4,095
13	₹500	₹4,096	₹6,500	₹8,191
15	₹500	₹16,384	₹7,500	₹32,767

• Bela's weekly amount first beats Arjun's in week 10 (₹512).
• Bela's total first beats Arjun's in week 13 (₹8,191 against ₹6,500).
• By week 20 Bela would be saving ₹524,288 in one week. (Her parents might have something to say about that!)

Adding the same amount each week (arithmetic) grows steadily. Doubling (geometric) starts slowly and then explodes. This is why scientists worry about anything that doubles, such as an infection spreading or a debt with high interest.

Mathematicians love the question “Is it always true?”. For any statement about numbers there are three possibilities:

• Always true: it works for every number you could try. You need a reason (an argument or proof) to be sure, because you can never try every number.
• Sometimes true: it works for some numbers and not others. Find one example where it works and one counterexample where it doesn't.
• Never true: it fails for every number. Again, you need a reason.

Testing examples is the way to start. Explaining is the way to finish.

Here is a warning every pattern detective needs. Look at 1, 2, 4, … and you will probably say the next term is 8 (doubling). But these rules all start 1, 2, 4:

• Double each time: 1, 2, 4, 8, 16, 32, …
• Add 1, then 2, then 3, …: 1, 2, 4, 7, 11, 16, …
• Add the two terms before, plus 1: 1, 2, 4, 7, 12, 20, …

Three different rules, three different futures, all from the same three numbers. And in the Deepen layer you will meet a real geometry problem, cutting a circle into pieces, whose answers go 1, 2, 4, 8, 16 and then, astonishingly, 31.

So when you find a rule, ask yourself two questions: does it fit every term I have? and do I know why it should keep working? The matchstick rules pass both tests, because you can see why each square adds 3 sticks. A rule guessed from a few numbers only passes the first.