Write 33 as the sum of 3 perfect squares
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25+4+4 is your answer .......
Answer:
iven that humans have been studying numbers for thousands of years, you might think we know everything about the number 3. But mathematicians recently discovered something new about 3: a third way to express it as the sum of three cubes. Expressing a number as the sum of three perfect cubes is a surprisingly interesting problem. It’s easy to show that most numbers can’t be written as one cube or the sum of two cubes, but it’s conjectured that most numbers can in fact be written as the sum of three cubes. Finding those three cubes, however, can be quite a challenge.
For example, we knew we could write 3 as 1³ + 1³ + 1³ and also as 4³ + 4³ + (−5)³, but for over 60 years mathematicians wondered if there was another way. This past September, Andrew Booker and Andrew Sutherland finally found a third solution:
3 = 569,936,821,221,962,380,720³ + (−569,936,821,113,563,493,509)³ + (−472,715,493,453,327,032)³
(If you want to check, don’t bother grabbing your calculator: Most aren’t built to remember this many digits. But WolframAlpha can handle it.)
In finding this new solution for 3, the mathematicians used techniques developed earlier this year, when Booker found the first-ever sum of three cubes for the number 33. But why did these breakthroughs take so long? Well, in the hunt for the right cubes, there is a lot of territory to cover, and there are few clues to lead us where we want to go. So the trick is to find smarter ways to search. To get a sense of the challenge and the solution, let’s start with a simpler question: How can we write 33 as a sum of three integers?
We can write 33 = 19 + 6 + 8, or 33 = 11 + 11 + 11, or 33 = 31 + 1 + 1. We can use negative numbers too, so we can write 33 = 35 + (−1) + (−1). There are infinitely many ways we can do this, since we can always increase one or two of the numbers and decrease another to compensate, so that 33 = 36 + (−1) + (−2), 33 = 100 + 41 + (−108), and so on.
What about writing 33 as a sum of three squares? We would need to find three “perfect squares” — numbers that are equal to an integer times itself, like 1 = 12, 9 = 32, and 64 = 82 — that add up to 33. After playing around, you might find that 33 = 42 + 42 + 12 and 33 = 52 + 22 + 22. Are there any more? Not really. You could replace a 4 with −4 and still get 33 = (-4)2 + 42 + 12, giving us a few different ways to write our solutions, but however you count them, there are only a handful of ways to write 33 as a sum of three squares.
That’s because when summing squares we don’t have the same flexibility we have when summing integers. We have fewer choices, and, more importantly, adding will only ever increase our sum. This is because perfect squares are never negative: Squaring a positive or negative integer always results in a positive integer.
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The squares are more restrictive, but something good comes from those restrictions: Our search space is “bounded.” In trying to find three squares that sum to 33, we can’t use any number whose square is bigger than 33, because once our sum of squares exceeds 33, there’s no way to decrease it. This means we only have to consider combinations of 0², 1², 2², 3², 4² and 5² (we’ll ignore their negative counterparts, which don’t really add anything new).
With only six options for each of our three squares, we have fewer than 6 × 6 × 6 = 216 ways that three squares could possibly sum to 33. That’s a small enough list to allow us to check each possibility and make sure we didn’t miss anything.
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