🧮 The maths magician

The answers, and why

1. What maths makes The Magic Number 1089 work? Place value, and reversing a three-digit number. A three-digit number minus its own reverse is always a multiple of 99: 198, 297, 396 and so on up to 891. Add any one of those to its own reverse and you get 1089. The rule about the end digits is just there to stop the difference being 0.

2. What maths makes The Secret Number Cards work? Binary: every number built from 1, 2, 4, 8, 16, 32. Every number can be built out of 1, 2, 4, 8, 16 and 32 in exactly one way, which is how computers count. A card holds every number that needs its first number to be built, so handing back the right cards is the same as telling you which pieces the number is made of. Adding them puts it back together.

3. What maths makes The Lightning Calendar work? The nine times table, and the middle of a square. In any three-by-three block, the numbers sit evenly around the middle one: whatever you take off on one side you add on the other. Pair them up across the middle and every pair is twice the middle number, so the nine of them come to nine times the middle. That is true of any calendar, any month.

4. What maths makes The Heads or Tails Mind Reader work? Odd and even, and what one change does to a count. Turning one coin over always changes the number of heads by exactly one, up or down. So the odd-or-even of that count flips with every single turn, and after an even number of flips it is back where it started. Knowing what it should be, and seeing what it is, leaves only the covered coin to explain the difference.

5. What maths makes The Birthday Mind Reader work? Multiplying through brackets, and a constant left over. Every step multiplies what is already there and adds a little more, so the day ends up multiplied by a hundred and the month is added on the end untouched. The spare bits add up to 165 every single time, whoever is playing, so taking 165 away leaves the day in the hundreds and the month in the units.

6. What maths makes The Answer Is Always 3 work? Doing and undoing: inverse operations. Doubling a number and adding 6 and halving it leaves their number plus 3. Taking their own number away leaves 3, and it does that whatever they started with, including a huge number or a fraction.

7. What maths makes The X-Ray Dice work? Opposite faces of a die always add to seven. Every die is made so that opposite faces add up to 7: 1 with 6, 2 with 5, 3 with 4. In a tower of three, five faces are hidden and one is on top, which is exactly three opposite pairs (21) minus the face you can see.

8. What maths makes The Calculator Prediction work? Factors, and why 12345679 × 63 does that. 12,345,679 × 9 = 111,111,111. Multiplying by nine lots of a digit is the same as multiplying by 9 and then by the digit, so the nine ones all become that digit instead.