Maths problem
#1
Posted 26 July 2013  10:13 PM
Given three unknown square numbers add up to 109, what are these numbers?
The conditions are that these numbers are all not equal, and they are whole numbers.
Is there a systematic, methodical way to determine these numbers, without resorting to "guess and check"?
#2
Posted 26 July 2013  10:33 PM
Calling the maths experts:
Given three unknown square numbers add up to 109, what are these numbers?
The conditions are that these numbers are all not equal, and they are whole numbers.
Is there a systematic, methodical way to determine these numbers, without resorting to "guess and check"?
A solution set (3,6,8) is quickly found by observing that 109 = 100 + 9 = 10^2 + 3^2.
10^2 = 5^2.2^2, and since (3,4,5) is a well known primitive Pythagorean triple, we can quickly deduce that the other numbers are 2*3 = 6 and 2*4 = 8.
Proving uniqueness is more challenging, and I'm still working on it.
If you want a more systematic method than this, I wager it'll be difficult. This is called a quadratic Diophantine equation, and the theory here is quite deep, involving Pelllike equations and whatnot. Quite cheem.
#6
Posted 27 July 2013  12:17 AM
The equation can be represented by a^2 + b^2 + c^2 = 109. The RHS (right hand side) is odd (it's also prime, but that's not that relevant here).
Clearly, a, b and c can't all be even. If exactly two of them are odd, they will sum to an even number, which when added to the last even number, will give an even number (which clearly can't be 109). Therefore at least one of a,b and c must be odd. That admits of 2 possibilities, either the LHS is odd + odd + odd or odd + even + even.
Let's eliminate one of these two possibilities. Observe that 109 is of the form 4k + 1 where k is an integer (simply put, this means one more than a multiple of four).
It can quite easily be proven that a perfect square can only be of the form 4s (if even) or 4s+1 (if odd), where s is an integer. Since we know that at least one of a^2, b^2 and c^2 is odd, let's subtract that from the RHS. With no loss of generality , we can say that a^2 is odd.
So we have: b^2 + c^2 = 109a^2, where a^2 is odd (implying a is odd). Since a has to be of the form 4s + 1, and 109 = 4k + 1, we get:
b^2 + c^2 = 4(k  s)
So b^2 + c^2 is a multiple of 4. Clearly b and c are both even, and we can now express b = 2m, c = 2n, giving:
(2m)^2 + (2n)^2 = 4(ks)
Hence,
m^2 + n^2 = k  s
Recall that k = 27 because 109 = 4*27 + 1. So we're left with the much simpler equation:
m^2 + n^2 = 27  s
Can we go eliminate further possibilities? Yes we can. Remember that we originally defined s by c^2 = 4s + 1. Let's rearrange that:
c^2  1 = 4s
(c+1)(c1) = 4s
s = (1/4)*(c+1)(c1)
Clearly, s has to be a natural number. The only way that can happen is for the product (c+1)(c1) to be divisible by 4, which implies that both factors are even. So c  1 = 2u, c + 1 = 2(u+1) giving s = u*(u+1). Therefore, s has to be the productive of consecutive natural numbers!
That makes things really simple when we do the final elimination. We now have to find natural numbers m and n such that m^2 + n^2 = 27  s, where s is itself the product of two consecutive natural numbers.
The only values of s that can be admitted are clearly: 1*2, 2*3, 3*4 and 4*5, since anything higher gives a negative value on the RHS.
So we have to consider solutions to:
m^2 + n^2 = 25
m^2 + n^2 = 21
m^2 + n^2 = 15
m^2 + n^2 = 7
The second to fourth equations can be solved by subtracting all possible values of n^2 from the RHS, e.g. for the second equation:
n^2 can be 1, 4, 6, 9, 16 but no higher. It's quickly seen that none of these values can give a valid m^2.
Ditto for equations 3 (test n^2 = 1, 4, 9) and 4 (test only n^2 = 1, 4).
The first equation, when similarly exhaustively tested (for n^2 = 1, 4, 9, 16), yields a valid solution for m^2 only for n^2 = 9 (or n = 3) and n^2 = 16 (or n = 4). In this case, m^2 = 2516 = 9 or 259 = 16 respectively, so either (m = 3, n=4) or (m=4, n=3). These are basically interchangeable solutions.
Therefore m = 3, n = 4 and s = 2 OR m = 4, n = 3, s = 2 gives the only valid solution sets, which corresponds to a^2 = 4*2 + 1 = 9 (=3^2), b^2 = 4*3^2 = 36 (=6^2) and c^2 = 4*4^2 = 64 (=8^2) or a^2 = 9, b^2 = 64, c^2 = 36. As mentioned, these two are essentially redundant and interchangeable.
Thus (3,6,8) is proven to be the only valid unique solution set for natural a, b and c. (QED)
[Note: Technically, zero is a whole number (integer), so (3,0,10) (keeping the order as above) is also a valid solution to the original problem if there is no requirement for a, b and c to all be natural numbers (which excludes zero). This solution occurs when you consider the equation m^2 + n^2 = 25 and allow for n^2 = 25, which gives m^2 = 0. So if this is allowed, there are, in fact, two unique solutions in the integers, namely: (3,6,8) and (0,3,10) when the solutions are rearranged in ascending order].
Edited by Turboflat4, 27 July 2013  12:28 AM.
#7
Posted 27 July 2013  12:47 AM
A solution set (3,6,8) is quickly found by observing that 109 = 100 + 9 = 10^2 + 3^2.
10^2 = 5^2.2^2, and since (3,4,5) is a well known primitive Pythagorean triple, we can quickly deduce that the other numbers are 2*3 = 6 and 2*4 = 8.
Proving uniqueness is more challenging, and I'm still working on it.
If you want a more systematic method than this, I wager it'll be difficult. This is called a quadratic Diophantine equation, and the theory here is quite deep, involving Pelllike equations and whatnot. Quite cheem.
knn,
I study so many years, also never see anything more cheem than what you have just typed....
heng after Air level no need to do math already
Edited by Sk65, 27 July 2013  12:49 AM.
#8
Posted 27 July 2013  12:59 AM
You won book prizes and you can solve super cheem math question.
I have always worshipped the tablewiper (wipe tables also can drive BMW and stay in big bungalow, better than me, everyday wipe kachen, ie. cleaning up mess created by others, and still money not enough), I think I have found a new idol.
This is impressive, I am now very curious to know what you do for a living.
#10
Posted 27 July 2013  01:46 AM
Calling the maths experts:
Given three unknown square numbers add up to 109, what are these numbers?
The conditions are that these numbers are all not equal, and they are whole numbers.
Is there a systematic, methodical way to determine these numbers, without resorting to "guess and check"?
Very easy,
1 sq =1
2 sq =4
3 sq =9
4 sq =16
5 sq =25
6 sq =36
7 sq =49
8 sq =64
9 sq =81
10 sq =100
So it can't be 1, cause the other 2 cannot be 108.
It can't be 2, because the other 2 cannot be 105.
It is 3 because the other 2 is 6 and 8. 3sq is 9+ 6 sq is 36 +8 sq is 64.
So the only answer got to be
#11
Posted 27 July 2013  02:53 AM
OK, I've found a way to prove uniqueness of the result for the squares of three natural numbers a, b and c. Not the most elegant, but it'll do. It also allows a slightly more systematic solution of the equation by elementary means.
Woah, this is so cheem leh.. Thought I'd see something do to with algebra, but when I see your reply, my eyes go already..
就能享受别人所不能享受的一切.
#14
Posted 27 July 2013  07:02 AM
Very easy,
1 sq =1
2 sq =4
3 sq =9
4 sq =16
5 sq =25
6 sq =36
7 sq =49
8 sq =64
9 sq =81
10 sq =100
So it can't be 1, cause the other 2 cannot be 108.
It can't be 2, because the other 2 cannot be 105.
It is 3 because the other 2 is 6 and 8. 3sq is 9+ 6 sq is 36 +8 sq is 64.
So the only answer got to be
Short, simple and elegant. Best post here
#15
Posted 27 July 2013  07:05 AM
Calling the maths experts:
Given three unknown square numbers add up to 109, what are these numbers?
The conditions are that these numbers are all not equal, and they are whole numbers.
Is there a systematic, methodical way to determine these numbers, without resorting to "guess and check"?
This sounds very GMAT
#16
Posted 27 July 2013  07:11 AM
Very easy,
1 sq =1
2 sq =4
3 sq =9
4 sq =16
5 sq =25
6 sq =36
7 sq =49
8 sq =64
9 sq =81
10 sq =100
So it can't be 1, cause the other 2 cannot be 108.
It can't be 2, because the other 2 cannot be 105.
It is 3 because the other 2 is 6 and 8. 3sq is 9+ 6 sq is 36 +8 sq is 64.
So the only answer got to be
Bro, criteria no guess and check liao. Answer correct but working wrong. No marks awarded.
#17
Posted 27 July 2013  07:47 AM
Very easy,
1 sq =1
2 sq =4
3 sq =9
4 sq =16
5 sq =25
6 sq =36
7 sq =49
8 sq =64
9 sq =81
10 sq =100
So it can't be 1, cause the other 2 cannot be 108.
It can't be 2, because the other 2 cannot be 105.
It is 3 because the other 2 is 6 and 8. 3sq is 9+ 6 sq is 36 +8 sq is 64.
So the only answer got to be
I'm sorry  but I have to say that it's not clear what you're doing here.
Why can't the sum of the other 2 be equal to 108 or 105?
You're correct that the max value of any of three can only be 10. But even if so, the sum of squares of two numbers both less than ten can still add up to numbers like 106 (5^2 + 9^2), so how exactly have you eliminated these possibilities?
Basically if you're using completely naive "guess and check". you actually have to subtract perfect squares from 109, then test which of those are expressible as the sum of two squares themselves. This is tedious, and I believe the OP does not want it done this way.
But if you had another insight that allows you to "automatically" exclude the possibility that the sum of the other two squares can be those numbers, I'm curious to know. Always looking for a more elegant method.
Edited by Turboflat4, 27 July 2013  07:52 AM.
#18
Posted 27 July 2013  07:52 AM
You won book prizes and you can solve super cheem math question.
I have always worshipped the tablewiper (wipe tables also can drive BMW and stay in big bungalow, better than me, everyday wipe kachen, ie. cleaning up mess created by others, and still money not enough), I think I have found a new idol.
This is impressive, I am now very curious to know what you do for a living.
Please don't idolise me lah bro. I'm just a humble yisheng, and an especially humble one at that  definitely no surgeon earning the big bucks.
Continue to idolise T2, after all, I do too!
I've always been keen on Math/Physics  had a PSC OMS scholarship to do EE at Caltech, but turned it down to do Medicine here so that I obey father's wishes and be close to home. Bit late to regret now.
#19
Posted 27 July 2013  07:55 AM
As I said, if a completely rigorous solution is sought, the theory is very deep and frankly, the working will be even more involved, and not worth the trouble. Not unless the RHS is a ridiculously big number like 123456789, which makes "guess and check" without a computer almost impossible. And if allowed access to a PC, it's still faster to write a simple program to slog through a reduced number of possibilities than try to attack the problem head on with the theory of quadratic Diophantines.
Edited by Turboflat4, 27 July 2013  07:57 AM.
#20
Posted 27 July 2013  09:06 AM
Please don't idolise me lah bro. I'm just a humble yisheng, and an especially humble one at that  definitely no surgeon earning the big bucks.
Continue to idolise T2, after all, I do too!
I've always been keen on Math/Physics  had a PSC OMS scholarship to do EE at Caltech, but turned it down to do Medicine here so that I obey father's wishes and be close to home. Bit late to regret now.
Wow, noble profession and filial son.
Yes, T2 has quite a following in MCF, got 2 camps. For the record, I am in the good camp. Haha....

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