Number Magic

Before you even pick your number, I seal my prediction in an envelope. Choose a three-digit number whose first and last digits are at least 2 apart, reverse it, subtract the smaller from the bigger, then add that answer to its own reverse. Open the envelope: how could I have known?

Think of a secret number and don't tell anyone. Double it, add 10, cut it in half, then take away the number you first thought of. I can tell you what's left without asking: how could I possibly know?

Take any number with a few digits, scramble its digits to make a new number, and subtract the smaller one from the bigger one. Whatever you started with, the answer always divides exactly by one special number. Which one is it?

Pick a four-digit number whose digits aren't all the same, then subtract its digits in smallest-first order from the same digits in biggest-first order. Do the same thing to your answer, again and again, round after round. What do you think will happen?

Here's a game: if your number is even, halve it; if it's odd, triple it and add 1; then keep going. Starting at 6 you get 6, 3, 10, 5, 16, 8, 4, 2, 1. Does every starting number end up at 1?

Your friend says she can multiply any two-digit number by 11 faster than you can type it into a calculator. Her secret hides in these answers: 45 × 11 = 495, 63 × 11 = 693, 26 × 11 = 286. Can you spot it?

Start with a one in the middle of some squared paper and write the counting numbers round and round it in a spiral. Then shade every prime number, the ones that divide only by one and themselves. When the spiral gets really big, what picture do you think the shaded squares make?

Count the spirals of seeds in a sunflower, or the bumpy rows on a pineapple, and you keep finding the same list of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34 … What comes next?

Some numbers are 'perfect': add up all the smaller numbers that divide into them evenly, and you get the number back exactly. Six is one, because 1 + 2 + 3 = 6. The next perfect number is hiding somewhere between 20 and 30: can you find it?

Three friends share a hotel room and pay 30 dollars between them. The clerk realises the room costs only 25 and sends the bellboy back with 5, but the bellboy quietly keeps 2 and gives each friend 1 back. So each friend paid 9, and 3 × 9 = 27, plus the bellboy's 2 makes 29: where did the missing dollar go?