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Cracking Combinations is a puzzle in Professor Layton and the Azran Legacy. The puzzle must be solved in order to progress the story.

Puzzle

The police have received a warning from a legendary safe-cracker about his next crime:

"Tonight, I plan to break into the bank and take the treasure from one of the safes numbered 1 to 60. I will steal from the one safe whose number cannot be made by adding together any combination of these: 0, 1, 2, 4, 7, 15, 31. (Using each one no more than once per combination, that is.)"

Which safe should the police keep an extra close eye on tonight?

Hints




Click a Tab to reveal the Hint.

Here are a couple of safe numbers you can create by adding together some of the numbers the thief gave:

1 + 2 = 3 1 + 4 = 5

See if you can create some other safe numbers by adding together the numbers that are listed in the warning note. First, try making the numbers 1 - 14.

If you've followed Hint 1 and used various combinations of the numbers that the note lists between 0 and 7 to create every safe number between 1 and 14, the next step is to use the 15 to create even bigger safe numbers!

Following on from Hint 2, you should now have been able to make every number from 1 to 29 using combinations of all the numbers that the note lists from 0 to 15.

Now, before you start trying to create even higher safe numbers, think about all the combinations you've used to far. That should reduce the amount of calculation you need to do.

Simply adding together various combinations of 0, 1, 2, 4, 7, 15 and 31 allows you to create every number from 1-29 and from 31-60.

Once you've worked that out, the answer to this puzzle should be obvious.


Solution

Incorrect

Too bad.

Read the legendary safe-cracker's warning one more time, and then give it another go.

Correct

Cracking!

The legendary safe-cracker is targeting safe number 30. The numbers 1-29 and 31-60 can all be made by adding together various combinations of 0, 1, 2, 4, 7. 15, and 31.

30 is the only number between 1 and 60 that cannot be made from these numbers.

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