HAPPY NUMBERS

1. HAPPY NUMBERS

2. A group of students are taken on a camp. • They set up camp in a clearing in the bush. • The students are given permission to go and have a wander nearby in the bush, but be back in time for dinner.

3. The happy students are the ones who can get back to camp for dinner • The unhappy students are the ones who find they are going around in circles in the bush and not able to get back to camp for dinner

4. Getting back to camp is getting back to 1. • Any numbers which are on a path back to 1 are happy. • Begin with a 2 digit number, • square each digit, then add together • repeat until you are on the happy path, or have gone around in a circle.

5. 22 • + 32 • = 13 23 → 13 → • 12 • + 32 • = 10 10 → • 12 • + 02 • = 1 • Hence 23, 13, 10 and 1 are all happy

6. 24 → • 22 + 42 = 20 20 → • 22 + 02 = 4 4 → • 42= 16 16 → • 12 + 62 = 37 37 → • 32 + 72 = 58 58 → • 52 + 82 = 89 89 → • 82 + 92 = 145 145 → • 12 + 42 + 52 = 42 42 → • 42 + 22 = 20 We have met 20 before; we are going in a circle; 24, 20, 4, 16, 37, 58, 89, 145 and 42 are all unhappy