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#172 Four got it

Emma Neesha is a forgetful sort, and she has just locked herself out of her house with her keys still inside.

Not to worry, she installed keypads on both her house and car for such occasions. Unfortunately, she has also forgotten the four-digit code to her house.

This shouldn鈥檛 be a problem, because she keeps a code clue in her wallet. It says: 鈥淔our times the car鈥檚 four-digit code.鈥�

Well, that is fine, but she has forgotten that one too. Fortunately, Emma is prepared: she has a clue written down for the car鈥檚 code as well. Pulling that one out, it reads: 鈥淭he reverse of the house code.鈥� Oh dear.

Can you help?

#171 The magic number bracelet

Solution

The beads are 1, 3, 10, 2, 5.

There are 21 combinations you can make with five different numbers (the five individual numbers, five adjacent pairs, threes and fours, and the whole set of five numbers). This means there must be exactly one way to make each number from 1 to 21. To make 1 and 2, the bead numbers must include numbers 1 and 2. No bead can be >11 as 1 + 2 +3 + 4 + 11 = 21. Three adjacent beads add to 17, so the other two add to four and must be 1 and 3, so there is no 4 bead.

That leaves two possible arrangements: 1, 3 , 2 , *, *, meaning there can鈥檛 be a 5 or 6, leaving 1, 3, 2 (7/8), but no combination works. Or 1, 3, *, 2, *, so to make 5 there needs to be a 5 bead, leading to 1, 3, 10, 2, 5.