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Rectangle numbers

Number theory · level 1, lesson 5 of 6

A question to start

You have 6 counters. You can lay them out in a neat rectangle: 2 rows of 3. Now take one more counter, so you have 7. Can you make a neat rectangle with 7?

Try it with real counters, coins or buttons before you read on. Move them around as much as you like.

Here is the surprise: it cannot be done. 7 counters can only make one long line. And 7 is not the only number like this. Some numbers love to make rectangles, and some numbers refuse. In this lesson you will find out which numbers up to 12 are which, and you will meet some of the most important numbers in all of mathematics.

The idea

What a rectangle of counters is

A rectangle of counters is a neat block of counters with:

  • rows going across, all the same length, and
  • at least 2 rows, with at least 2 counters in each row.

There are no gaps, and no counter sticks out. If you put a frame around it, the frame would be a rectangle, the shape you know from doors and books.

Here are 6 counters as a rectangle.

2 rows of 3 3 rows of 2 turn it

On the left: 2 rows of 3. On the right, the same counters turned on their side: 3 rows of 2. Turning a rectangle does not make a new one. It is the same rectangle, just looked at from another side, like turning a book around on the table. So we count it once and call it "2 by 3".

A number that can make at least one rectangle is a rectangle number. 6 is a rectangle number.

Why 7 refuses

Let us try to build a rectangle with 7 counters, one row size at a time.

2 rows: 1 left 3 rows: 1 left a line of 7
  • 2 rows? Deal 7 counters into 2 rows. That is sharing between two friends, and 7 is odd. So 1 is left over. No rectangle.
  • 3 rows? Deal into 3 rows: each row gets 2, and 1 is left over. No rectangle.
  • 4 rows? You need at least 2 in each row, so 4 rows need at least 8 counters. But we only have 7. No.
  • More rows? Even more counters needed. No.

So 7 counters can only make one line: 1 row of 7. We call 7 a line-only number.

Notice the last step. You do not have to try every number of rows for ever. Once the rows need more counters than you have, you can stop.

Dealing into rows is sharing

Did you notice? Putting counters into 2 rows is sharing between 2 friends. Putting counters into 3 rows is sharing between 3 friends, and so on. Each row is one friend's share.

  • If the sharing comes out with nothing left over, the rows are all the same length, and you have a rectangle.
  • If something is left over, the leftover sticks out, and it is not a rectangle.

So "Is 7 a rectangle number?" is the same question as "Can 7 counters be shared fairly among 2, or 3, or more friends, with nothing left?" For 7, the answer is no, every time.

Every number up to 12

Let us do this for every number from 2 to 12. Here are all the rectangles we can find.

4: 2 by 2 6: 2 by 3 8: 2 by 4 9: 3 by 3 10: 2 by 5 12: 2 by 6 12: 3 by 4

And here is the whole list.

Number Rectangles Kind
2 none (only 1 row of 2) line-only
3 none line-only
4 2 by 2 rectangle
5 none line-only
6 2 by 3 rectangle
7 none line-only
8 2 by 4 rectangle
9 3 by 3 rectangle
10 2 by 5 rectangle
11 none line-only
12 2 by 6, and 3 by 4 rectangle (two ways!)

The line-only numbers up to 12 are 2, 3, 5, 7 and 11.

A few things to notice:

  • 4 and 9 make squares. 2 by 2 and 3 by 3 have the same number of rows as counters in a row. A square is a special rectangle, so they count.
  • 12 makes two different rectangles, 2 by 6 and 3 by 4. That is why egg boxes often hold 12 eggs (2 rows of 6), and muffin trays often hold 12 muffins (3 rows of 4). 12 is a very friendly number.
  • Every even number from 4 up is a rectangle number, because you can always put it in 2 rows (it shares fairly between two). So 2 is the only even line-only number. 2 has to be a line, because 2 rows of 1 is not allowed (each row needs at least 2).
  • Odd numbers can be rectangles too. 9 is 3 by 3. (Later you will find 15 is 3 by 5.)

What about 1?

1 counter is just one counter. It is not a rectangle, and it is not really a line either. So 1 is left out of the game. It is neither a rectangle number nor a line-only number. Mathematicians treat 1 as special, and you will see why in level 2.

The line-only numbers have a famous name

The line-only numbers, 2, 3, 5, 7, 11, 13, 17, 19, ..., are called prime numbers by mathematicians. They are some of the most studied numbers in all of math, and there are still questions about them that nobody has been able to answer. You will meet them properly in Primes and composites in level 2.

A rectangle on Mars

The rovers Curiosity and Perseverance, which drive on Mars, each have 6 wheels: 3 on the left side and 3 on the right. Seen from above, the wheels make a rectangle: 2 rows of 3.

6 wheels: 2 rows of 3

Worked examples

Example 1. Is 10 a rectangle number?

  1. Try 2 rows. 10 is even, so it shares fairly between two: 5 in each row.
  2. That is a rectangle: 2 by 5. So yes, 10 is a rectangle number.
  3. Are there more? Try 3 rows: 3 in each row and 1 left over. No. Try 4 rows: 2 in each row and 2 left over. No. 5 rows of 2 is just 2 by 5 turned around. So 10 has exactly one rectangle.

Example 2. Is 11 a rectangle number?

  1. 2 rows: 11 is odd, so 1 is left over. No.
  2. 3 rows: 3 in each row is 9, and 2 are left over. No.
  3. 4 rows: 2 in each row is 8, and 3 are left over. No.
  4. 5 rows: 2 in each row is 10, and 1 is left over. No.
  5. 6 rows would need at least 12 counters. Too many. Stop.
  6. So 11 is line-only.

Example 3. Every rectangle for 12.

  1. 2 rows: 6 in each. 2 by 6.
  2. 3 rows: 4 in each. 3 by 4.
  3. 4 rows: 3 in each. But 4 by 3 is 3 by 4 turned around. Already found!
  4. 5 rows: 2 in each row is 10, and 2 are left. No.
  5. 6 rows: 2 in each. That is 2 by 6 turned around. Already found.
  6. So 12 makes two rectangles: 2 by 6 and 3 by 4.

Notice step 3: once you start finding the same rectangles turned around, there is nothing new to find.

Try it

  1. Make a rectangle with 8 counters. How many rows? How many in each row?
  2. Is 5 a rectangle number? Try 2 rows and 3 rows.
  3. Draw the square you can make with 9 counters.
  4. Which numbers up to 12 are line-only?
  5. Is 14 a rectangle number? Is 13?
  6. Is 15 a rectangle number? Draw it.
  7. A muffin tray holds 3 rows of 4. How many muffins is that? Can you think of a different rectangle with the same number of muffins?
  8. Tia says: "Every odd number is line-only." Find a number that shows she is wrong.
Answers
  1. 2 rows of 4 (or, turned around, 4 rows of 2). It is the same rectangle: 2 by 4.
  2. No. 2 rows: 2 in each and 1 left over. 3 rows: each row needs at least 2, so 3 rows need at least 6 counters, and we only have 5. So 5 is line-only.
  3. 3 rows of 3. Draw three dots, then three under them, then three more under those.
  4. 2, 3, 5, 7 and 11.
  5. 14 is a rectangle number: it is even, so 2 rows of 7. 13 is line-only: 2 rows of 6 leaves 1, 3 rows of 4 leaves 1, 4 rows of 3 leaves 1, 5 rows of 2 leaves 3, 6 rows of 2 leaves 1, and 7 rows would need at least 14 counters, which is too many.
  6. Yes: 3 rows of 5. (15 is odd, so 2 rows does not work, but 3 rows does.)
  7. 3 rows of 4 is 12 muffins (count by 4s: 4, 8, 12). A different rectangle with 12 is 2 rows of 6, like an egg box.
  8. 9 (3 by 3) or 15 (3 by 5). Both are odd and both make rectangles. Being odd only means 2 rows will not work; 3 rows (or more) might.

Common mistake

Counting a turned rectangle as new. 3 by 4 and 4 by 3 are the same rectangle, turned. Count it once.

Forgetting that a square is a rectangle. 4 counters make a 2 by 2 square. A square has rows all the same length and is a special rectangle, so 4 is a rectangle number.

Calling a number line-only too soon. 9 does not work with 2 rows, but it is not line-only: try 3 rows. Keep trying more rows until the rows would need more counters than you have.

Where it leads