In 30 seconds
3Blue1Brown poses a counting problem about subsets of the integers from 1 to 2,000. A five-number example illustrates the question; the short clip points to a longer video for the solution.
- 1The questionCount subsets of {1, …, 2000} whose elements add up to a multiple of five.
- 2A smaller exampleAmong the 32 subsets of {1, 2, 3, 4, 5}, eight have a sum divisible by five.
- 3An exact countThe challenge is to find the precise answer for the larger set, not just estimate it.
Takeaways
- The problem asks for subsets whose element sums are divisible by five.
- The set {1, 2, 3, 4, 5} has 32 subsets, eight of which have sums divisible by five.
- The full set has 2^2000 subsets, making the exact count a much larger challenge.
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The puzzle and a small example
The speaker introduces the subset-sum question and illustrates it with five numbers.
How many subsets of {1, 2, 3, …, 2000} have elements that sum to a multiple of five? The question is easy to state, but finding an exact count is the challenge.
Start with the smaller set {1, 2, 3, 4, 5}. It has 2^5 = 32 subsets. If you add the elements in each subset and group the results by sum, eight have a sum divisible by five; the empty subset counts because its sum is zero.
The larger challenge
The exact count for 2,000 numbers requires a deeper technique, explained in the linked full video.
For the set through 2,000, there are 2^2000 subsets. A rough estimate might be tempting, but the problem asks for the exact answer. The speaker points to a surprising mathematical technique in the [full video](https://youtu.be/bOXCLR3Wric); this short clip does not give the solution.
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