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Square numbers

Patterns and sequences · level 3, lesson 2 of 6

A question to start

Add up the first ten odd numbers:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

Do it slowly on paper. The answer is a very round number: 100. Now try the first five odd numbers: 1 + 3 + 5 + 7 + 9 = 25. And the first three: 1 + 3 + 5 = 9.

100, 25, 9. Do you recognise them? They are \(10 \times 10\), \(5 \times 5\) and \(3 \times 3\). Is that luck, or is something making it happen? In this lesson you will see the reason in a picture of dots.

The idea

Numbers that make squares

Take some dots and try to arrange them in a perfect square: the same number of rows as columns, with no dots left over. Some numbers can do it and some cannot.

  • 9 dots make a square: 3 rows of 3.
  • 10 dots cannot. 3 rows of 3 leaves one dot over; 4 rows of 4 needs 16.

A number that makes a perfect square of dots is called a square number. A square with 4 dots along each side holds \(4 \times 4 = 16\) dots, so 16 is a square number. We say "4 squared is 16" and write it \(4^2 = 16\). The small raised 2 means "multiply the number by itself" (two copies of 4 multiplied together). It does not mean "times 2".

Side 1 2 3 4 5 6 7 8 9 10 11 12
Square number 1 4 9 16 25 36 49 64 81 100 121 144

The side is also the position in the sequence: the 7th square number is \(7 \times 7 = 49\). So you can jump straight to any square number: the 20th is \(20 \times 20 = 400\).

Growing one square into the next

Now draw the squares one inside the next, and colour each new layer.

1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 ...

Each new layer is shaped like a letter L that wraps around two sides of the old square. Count the dots in each L:

  • 1 (the very first dot)
  • 3 (to go from 1 to 4)
  • 5 (to go from 4 to 9)
  • 7 (to go from 9 to 16)

The L's are the odd numbers, in order. So the square of side 4 is built from 1 + 3 + 5 + 7, and that is why 1 + 3 + 5 + 7 = 16. The ten odd numbers in the opening question make ten L's, which wrap up into a 10 by 10 square: 100.

Why the next L is always the next odd number

Look closely at how a 4 by 4 square grows into a 5 by 5 square.

4 × 4 = 16 blue new row of 4 (green) new column of 4 (green) 1 corner dot (orange) 4 + 4 + 1 = 9, the 5th odd number 16 + 9 = 25 = 5 × 5

You add a row of 4 along the top, a column of 4 up the side, and one dot to fill the corner. That is \(4 + 4 + 1 = 9\).

The same thing happens at every size. To grow a square of side 6 into side 7, you add a row of 6, a column of 6 and a corner: \(6 + 6 + 1 = 13\). A number added to itself is always even, and one more than an even number is always odd. So every L is odd. And each L is exactly 2 more than the one before, because the row and the column each got 1 longer. That is why the L's go 1, 3, 5, 7, 9, ..., the odd numbers in order, with none missed.

The gap between two squares

There is a second way to read the same picture. Going from \(4 \times 4\) to \(5 \times 5\) we added \(4 + 4 + 1\), which we can also group as \(4 + 5\): the old side plus the new side. Check: \(25 - 16 = 9 = 4 + 5\). ✓

So the gap between two square numbers next to each other is the sum of their sides:

  • \(10 \times 10 = 100\) and \(11 \times 11 = 121\): the gap is \(10 + 11 = 21\). ✓
  • From \(12 \times 12 = 144\) you can get \(13 \times 13\) without multiplying: \(144 + 12 + 13 = 169\).

The last digit of a square

Look at the last digits of the squares: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, then 1, 4, 9, 6, 5, ... again. They repeat every ten. And some digits never appear at all: no square number ends in 2, 3, 7 or 8.

Why? The last digit of a product only depends on the last digits of the numbers you multiply. For example \(23 \times 23\) ends in the same digit as \(3 \times 3 = 9\), because the tens and hundreds cannot change the ones column. So every square ends in the same digit as one of \(0 \times 0, 1 \times 1, \ldots, 9 \times 9\), and those end in 0, 1, 4, 9, 6, 5, 6, 9, 4, 1. The digits 2, 3, 7 and 8 never show up. That gives a fast test: 4,357 cannot be a square number, without any multiplying.

(A number ending in 0, 1, 4, 5, 6 or 9 might be a square, but it might not: 11 ends in 1 and is not a square. The test can say "no", but it can never say "yes".)

Worked examples

1. Add 1 + 3 + 5 + 7 + 9 + 11.

  • Count the odd numbers: there are 6.
  • Six L's wrap into a 6 by 6 square, so the sum is \(6 \times 6 = 36\).
  • Check by adding: \(1 + 3 = 4\), \(+ 5 = 9\), \(+ 7 = 16\), \(+ 9 = 25\), \(+ 11 = 36\). ✓ (Notice that every running total is a square.)

2. Find \(13 \times 13\) from \(12 \times 12 = 144\).

  • The gap between the two squares is \(12 + 13 = 25\).
  • \(144 + 25 = 169\). So \(13 \times 13 = 169\).

3. Is 1000 a square number?

  • The last digit is 0, so the quick test cannot rule it out.
  • Look for nearby squares: \(30 \times 30 = 900\), \(31 \times 31 = 961\), \(32 \times 32 = 1024\).
  • 1000 falls between 961 and 1024, so no whole number squared makes it. Not a square.

4. A square patio. A patio is a square of 7 by 7 tiles. The owner wants a square of 8 by 8. How many new tiles?

  • New row of 7, new column of 7, one corner: \(7 + 7 + 1 = 15\).
  • Check: \(64 - 49 = 15\). ✓

Try it

  1. Write the next three square numbers after 49.
  2. Work out 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 without adding them one by one.
  3. \(20 \times 20 = 400\). Use the gap idea to find \(21 \times 21\).
  4. Can 2,348 be a square number? How can you tell so quickly?
  5. A square of tiles is 9 by 9. How many tiles must you add to make it 10 by 10?
  6. Which odd number do you add to 81 to get the next square number?
  7. List all the square numbers between 50 and 150.
  8. Work out 1 + 3 + 5 + ... + 39 (all the odd numbers up to 39). Hint: first find how many odd numbers there are.
Answers
  1. 64, 81, 100 (\(8 \times 8\), \(9 \times 9\), \(10 \times 10\)).
  2. There are 8 odd numbers, so they wrap into an 8 by 8 square: \(8 \times 8 = 64\).
  3. The gap is \(20 + 21 = 41\), so \(21 \times 21 = 400 + 41 = 441\).
  4. No. It ends in 8, and no square number ends in 2, 3, 7 or 8.
  5. A row of 9, a column of 9 and a corner: \(9 + 9 + 1 = 19\) tiles. (Check: \(100 - 81 = 19\).)
  6. \(81 = 9 \times 9\). The next square is \(10 \times 10 = 100\), and the gap is \(9 + 10 = 19\).
  7. 64, 81, 100, 121, 144. (\(7 \times 7 = 49\) is just below 50, and \(13 \times 13 = 169\) is above 150.)
  8. The odd numbers 1, 3, 5, ... are 1 less than the even numbers 2, 4, 6, ... So 39 is 1 less than 40, and 40 is the 20th even number. That means 39 is the 20th odd number, and there are 20 odd numbers in the sum. They wrap into a 20 by 20 square: \(20 \times 20 = 400\).

Common mistake

Squaring is not doubling. \(5^2\) means \(5 \times 5 = 25\), not \(5 \times 2 = 10\). The little 2 tells you how many fives to multiply, not what to multiply by. A picture helps: a square of side 5 has 25 dots, far more than 10.

Miscounting the odd numbers. In 1 + 3 + 5 + ... + 15, the last number is 15, but there are only 8 odd numbers, so the sum is \(8 \times 8\), not \(15 \times 15\). To count them, add 1 to the last odd number and halve: \((15 + 1) \div 2 = 8\).

Where it leads