Why would someone ever do that?
Once, on a rainy Friday, I went back from school, got home and had lunch. After that, I got really bored while watching TV, so I decided to build towers with my 504 wooden blocks. After some minutes, I realized that I was able to build a huge number of towers using my wooden blocks. So, in order to reduce this number, I only built towers that had the same number of blocks. I liked so much building those towers that I decided to show them to my brother. However, I only built sequences of up to 3 towers to show him.
Let a be the sum of the number of towers of every possible sequence that could have been made by me, and let b be the number of blocks that I used to build each tower of all the possible sequences that was shown to my brother. What is the value of a + b ?
Notes and Assumptions
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A sequence of towers is a sequence of a certain number of towers with the same number of wooden blocks. Every sequence made by me had a total number of 504 blocks, in other words, I used all the wooden blocks in every sequence.
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Consider that the space which I used to build the towers is infinite.
The answer is 2484.
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1 solution
The number 504 can be written as: 72x7, 36x14, 18x28, 9x56, 3x168, 2x252, 1x504, 4x126, 8x63, 6x84, 12x42 or 24x21.
When we write 504 = 72x7, we mean that you can build 7 towers of 72 blocks each, or 72 towers of 7 blocks each. These are two different sequences. The number of towers that were built after making these two sequences was 72 + 7, which is 79. So, to calculate a , we need to sum all these numbers: 72 + 7 + 36 + 14 + 18 + 28 + ... + 24 + 21, because, for each two sequences with the same numbers ( x blocks and y towers, or y blocks and x towers), the number of towers that are built is always the sum of the numbers ( x + y ). Hence, a = 1560 .
To calculate b , we need to know that, as you will only build sequences of up to 3 towers, you can build sequences with 1, 2 or 3 towers. So, when you build a sequence with 1 tower, you use 504 blocks in each tower (you only build one tower). When you build a sequence with 2 towers, you use 252 blocks to build each one. When you build a sequence with 3 towers, you use 168 blocks to build each one. Hence, b = 504 + 252 + 168 = 924 . Thus, a + b = 2484 .