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The Big-Five Blaster
Square any number ending in 5 in your head, even three-digit ones like 135 or 995. Spot the pattern first. The rule comes after you have found it yourself.
Look at these four squares
What do all four answers end in?
Now cover up the 25
Look at what is left in front: 110, 132, 156, 182. Compare it with the gold digits in each number. How is 110 related to 10? How is 132 related to 11?
You found the rule
Cover the 5. Multiply the number in front by the next number up. Then write 25 at the end.
Let’s see how 15 × 15 makes 225 with square tiles. Cut the 15 by 15 square at 10 and 5. That gives a 10 by 10 block, a 5 by 5 corner, and two strips of 10 by 5.
Now slide the bottom strip round to the right-hand side. The 10 by 10 block and the two strips make one long rectangle, 10 tall and 20 wide. The little corner is left over.
The rectangle is 10 × 20 = 200 tiles, and the corner adds 25. So 152 = 200 + 25 = 225. The 1 in front of the 5 became 1 × 2 = 2 hundreds, which is exactly the trick: the number in front times the next number up, then 25.
(10n + 5)2 = (10n + 5)(10n + 5) = 100n2 + 50n + 50n + 25 = 100n2 + 100n + 25 = 100 × n(n + 1) + 25
n(n + 1) is the number in front times the next number up. Multiplying by 100 leaves two empty places at the end, and 25 fills them. For n = 13: 13 × 14 = 182, then 18200, then 18225.
Eight questions, no calculator
Practice round done
Play Beat the Calculator
Challenge someone at home to a race. They choose any number from 105 to 195 that ends in 5 and say it out loud. They square it on a calculator, and you square it in your head. Whoever says the full answer first scores a point. Play five rounds. Most people win by round three.
Show-off move: once it feels easy, ask for 995. Nobody expects 9952 to take two seconds.
Ready for 43 × 47?
There is a sister trick for multiplying numbers like 43 × 47, and it grows straight out of what you just learned. I’ve put it in a free cheat sheet, together with a second trick for pairs like 47 × 53.
Subscribe to my Substack and the cheat sheet arrives in your welcome email.
Want all six squaring tricks?
This mission comes from Math Magic Missions Book 1: Squaring Secrets. The book has all six tricks, pictures that show why each one works, Bronze, Silver and Gold practice with a full answer key, and a printable certificate.