Corner by corner

About this video

Which numbers of dots can be laid out as a perfect square, with as many rows as there are dots in each row? This short animation builds them one after another: 1 dot, then 2 rows of 2, 3 rows of 3, 4 rows of 4 and 5 rows of 5. These are the square numbers, 1, 4, 9, 16 and 25: each one is a whole number multiplied by itself.

Most numbers are not square. Twelve dots make 3 rows of 4 or 2 rows of 6, but never a square, because the number of rows never matches the number in a row. Then the video builds the 5 by 5 square again, one corner-shaped strip at a time, each in its own colour. The strips have 1, 3, 5, 7 and 9 dots: the odd numbers, one after another. So 1 + 3 + 5 + 7 + 9 = 25, and adding the odd numbers in order always gives a square number. The picture shows why, and not just for these five: every square grows into the next by one more corner strip, two dots longer than the one before. This video goes with the Square Numbers lesson of Number Theory Level 1 on Knaph.

Transcript

1. 0:00Which numbers make a perfect square?
2. 0:04One, four, nine, sixteen, twenty-five. A number times itself.
3. 0:11Twelve makes rectangles. Three by four, two by six. Never a square.
4. 0:16Grow a square corner by corner. One, three, five, seven, nine.
5. 0:23Add the odd numbers and you get a square.