Digital Agent Blog | DECEMBER PUZZLER
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DECEMBER PUZZLER

DECEMBER PUZZLER

Digital Agent engineers solve complex puzzles for our customers every day. Can you solve the Puzzle we created for you?

 

Your spouse convinced you to spend your holiday budget on a fancy photographer this year. Your holiday cards look fabulous, but now your holiday stash is TIGHT! You cannot waste any more dough. You need to purchase the exact number of cards you intend to send. 

Cards only come in packs of 5 or 7, and you can buy any combination of the two. What is the largest quantity of cards that is not possible to get by combining packs of 5 and 7?

The natural way to approach this problem is process of elimination. If we can find combinations of card sets that give us a certain number of cards, we know that this number cannot be the answer. Our goal is to find a way to guarantee that above a certain total number of cards, every number is possible.

Since we can buy packs of 5, notice that we can just add 10 (two more packs of 5) to get any possible amount that ends in the same number. (For example, since we can have 7 cards, we can also have 17, 27, 37, etc). We can start ruling out a lot of possibilities using this method. We will call this the “+10 rule”

We can see by trial and error that there are numbers in the 10-19 range which are impossible (18, for example), so we want to check numbers in the 20-29 range to see if there are any impossible combinations bigger than 18.

20 = 5 + 5 + 5 + 5

21 = 7 + 7 + 7

22 = 5 + 5 + 7

23 = impossible

24 = 5 + 5 + 7 + 7

25 = 5 + 5 + 5 + 5 + 5

26 = 5 + 7 + 7 + 7

27 = 5 + 5 + 5 + 5 + 7

28 = 7 + 7 + 7 + 7

29 = 5 + 5 + 5 + 7 + 7

Since we can make 33 = 5 + 7 + 7 + 7 + 7, we know that every number bigger than 23 is now possible (by using the “+10 rule”). So the answer must be 23.

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