Six real numbers so that product of any five is the sixth one
Clash Royale CLAN TAG#URR8PPP
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One needs to choose six real numbers $x_1,x_2,cdots,x_6$ such that the product of any five of them is equal to other number. The number of such choices is
A) $3$
B) $33$
C) $63$
D) $93$
I believe this is not so hard problem but I got no clue to proceed. The work I did till now.
Say the numbers be $a,b,c,d,e, abcde$. Then, $bcdot ccdot dcdot ecdot abcde=a$ hence $bcde= +-1$.
Basically I couldn't even find a single occasion where such things occcur except all of these numbers being either $1$ or $-1$. Are these all cases?
combinatorics
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add a comment |
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One needs to choose six real numbers $x_1,x_2,cdots,x_6$ such that the product of any five of them is equal to other number. The number of such choices is
A) $3$
B) $33$
C) $63$
D) $93$
I believe this is not so hard problem but I got no clue to proceed. The work I did till now.
Say the numbers be $a,b,c,d,e, abcde$. Then, $bcdot ccdot dcdot ecdot abcde=a$ hence $bcde= +-1$.
Basically I couldn't even find a single occasion where such things occcur except all of these numbers being either $1$ or $-1$. Are these all cases?
combinatorics
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9
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They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
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– J.G.
Mar 4 at 7:38
11
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I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
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– lioness99a
Mar 4 at 11:13
16
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For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
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– Carl Witthoft
Mar 4 at 13:42
1
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Seems related to the knapsack problem.
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– EJoshuaS
Mar 5 at 16:48
2
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In a similar way to @J.G. they could all be 1.
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– Andrew McOlash
Mar 5 at 19:07
add a comment |
$begingroup$
One needs to choose six real numbers $x_1,x_2,cdots,x_6$ such that the product of any five of them is equal to other number. The number of such choices is
A) $3$
B) $33$
C) $63$
D) $93$
I believe this is not so hard problem but I got no clue to proceed. The work I did till now.
Say the numbers be $a,b,c,d,e, abcde$. Then, $bcdot ccdot dcdot ecdot abcde=a$ hence $bcde= +-1$.
Basically I couldn't even find a single occasion where such things occcur except all of these numbers being either $1$ or $-1$. Are these all cases?
combinatorics
$endgroup$
One needs to choose six real numbers $x_1,x_2,cdots,x_6$ such that the product of any five of them is equal to other number. The number of such choices is
A) $3$
B) $33$
C) $63$
D) $93$
I believe this is not so hard problem but I got no clue to proceed. The work I did till now.
Say the numbers be $a,b,c,d,e, abcde$. Then, $bcdot ccdot dcdot ecdot abcde=a$ hence $bcde= +-1$.
Basically I couldn't even find a single occasion where such things occcur except all of these numbers being either $1$ or $-1$. Are these all cases?
combinatorics
combinatorics
edited Mar 4 at 22:13
Ant
17.5k23074
17.5k23074
asked Mar 4 at 7:13
ChakSayantanChakSayantan
492511
492511
9
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They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
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– J.G.
Mar 4 at 7:38
11
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I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
$endgroup$
– lioness99a
Mar 4 at 11:13
16
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For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
$endgroup$
– Carl Witthoft
Mar 4 at 13:42
1
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Seems related to the knapsack problem.
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– EJoshuaS
Mar 5 at 16:48
2
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In a similar way to @J.G. they could all be 1.
$endgroup$
– Andrew McOlash
Mar 5 at 19:07
add a comment |
9
$begingroup$
They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
$endgroup$
– J.G.
Mar 4 at 7:38
11
$begingroup$
I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
$endgroup$
– lioness99a
Mar 4 at 11:13
16
$begingroup$
For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
$endgroup$
– Carl Witthoft
Mar 4 at 13:42
1
$begingroup$
Seems related to the knapsack problem.
$endgroup$
– EJoshuaS
Mar 5 at 16:48
2
$begingroup$
In a similar way to @J.G. they could all be 1.
$endgroup$
– Andrew McOlash
Mar 5 at 19:07
9
9
$begingroup$
They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
$endgroup$
– J.G.
Mar 4 at 7:38
$begingroup$
They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
$endgroup$
– J.G.
Mar 4 at 7:38
11
11
$begingroup$
I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
$endgroup$
– lioness99a
Mar 4 at 11:13
$begingroup$
I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
$endgroup$
– lioness99a
Mar 4 at 11:13
16
16
$begingroup$
For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
$endgroup$
– Carl Witthoft
Mar 4 at 13:42
$begingroup$
For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
$endgroup$
– Carl Witthoft
Mar 4 at 13:42
1
1
$begingroup$
Seems related to the knapsack problem.
$endgroup$
– EJoshuaS
Mar 5 at 16:48
$begingroup$
Seems related to the knapsack problem.
$endgroup$
– EJoshuaS
Mar 5 at 16:48
2
2
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In a similar way to @J.G. they could all be 1.
$endgroup$
– Andrew McOlash
Mar 5 at 19:07
$begingroup$
In a similar way to @J.G. they could all be 1.
$endgroup$
– Andrew McOlash
Mar 5 at 19:07
add a comment |
5 Answers
5
active
oldest
votes
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Well, if there are an even number of $1$'s and $-1$'s, then the property holds, so that's $$binom60+ binom62+ binom64+ binom66=32$$
And then there’s the case that they’re all zero. Thus, there are $33$ total cases.
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I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
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– ChakSayantan
Mar 4 at 7:47
109
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You have made no argument as to why this accounts for all cases...
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– Morgan Rogers
Mar 4 at 13:58
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As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
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– Clement C.
Mar 5 at 17:07
2
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@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
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– Zachary Hunter
Mar 6 at 1:08
1
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@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
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– IMil
Mar 6 at 1:13
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show 1 more comment
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Here's an argument which extends to $n$ real numbers quite nicely! We can start by noting that
$$x_1 (x_2x_3x_4x_5x_6) = x_1^2$$
And by commutativity, we get $x_i^2 = x_j^2$ which implies that all the magnitudes are equal.
Now if $L$ is the magnitude, we also must have $L = L^5$, and from this we conclude that $L=0$ or $1$.
Now we just have to go through the possibilities. If $L = 0$, then we have all $0$'s
If $L = 1$, then we have all $-1$'s and $1$'s. This configuration works iff the number of $-1$'s is even, as this would imply that
$$fracprod_i=1^6 x_i1 = 1 text and fracprod_i=1^6 x_i-1 = -1$$
Now we can count configurations. There will be
$$sum_i=0^3 binom62i = 2^5$$
possibilities. And finally, we have $1+32 = 33$.
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2
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the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
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– Zachary Hunter
Mar 4 at 17:08
3
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Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
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– Isaac Browne
Mar 4 at 21:03
2
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This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
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– Yakk
Mar 5 at 2:16
2
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@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
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– Isaac Browne
Mar 5 at 2:33
4
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It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
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– Erick Wong
Mar 5 at 2:43
add a comment |
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Just to make the solution more explicit, the product of all six values is the square of any one of them, so they're all $pm k$ for some $k$, and $k^6=k^2implies kin0,,1$. We get one $k=0$ solution and, because we require a non-negative product, one $k=1$ solution for each choice of six $pm 1$s with an even number of $-1$s. Famously, the numbers of even- and odd-sized subsets of a finite non-empty set are equal, and so the total number of solutions is $1+frac12cdot2^6$, i.e. B)
.
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add a comment |
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Even though the question stated real numbers, we can extend this to complex fairly easily.
Note we require
$$ exp( left( E - I) boldsymbolz right ) = exp(boldsymbolz) $$
where $exp(boldsymbolz) = boldsymbolx$ (the vector of numbers to multiply), $E$ is a matrix of ones and $I$ is the identity matrix.
Now,
$$ exp( left( E - 2I) boldsymbolz right ) = exp(2pi i boldsymbolk) $$
for some vector of integers $boldsymbolk$. Equating the powers and using $(E - 2I)^-1 = frac18(E - 4I)$, gives
$$ boldsymbolz = fracK pi i4 - pi i boldsymbolk$$
where $K = sum_i=1^6 (boldsymbolk)_i$ and hence
$$ boldsymbolx = exp left (fracK pi i4 right) cdot exp(pi i boldsymbolk)$$
The second term is clearly a vector of 1s and -1s.
The first term can take the following values:
If the total 1s in the second term are even i.e. (even number of even $k_i$),
- $1$
- $i$
- $-1$
- $-i$
If the total 1s in the second term are odd i.e. (odd number of even $k_i$),
- $frac1sqrt2 + frac1sqrt2i$
- $-frac1sqrt2 + frac1sqrt2i$
- $-(-frac1sqrt2 + frac1sqrt2i)$
- $-(frac1sqrt2 + frac1sqrt2i)$
Finally, consider the subsets of $A = -1, 1^6$, $A_E subset A$ containing an even number of 1s and $A_O subset A$ containing an odd number of 1s.
Clearly $A_E = -A_E$ (multiplying each element in the set by -1) and $A_O = - A_O$, hence the unique combinations of complex numbers is the union of disjoint sets
$$C = A_E cup iA_E cup (frac1sqrt2 + frac1sqrt2i) A_O cup (-frac1sqrt2 + frac1sqrt2i) A_O$$
As $|A_E| = |A_O| = 32$, and the above is a union of disjoint subsets, $|C| = 128$ and including the case where all values are $0$, we have the total complex combinations as $129$.
Note
To answer the question, we know that the only subset containing real values is $A_E$ so the total combinations in the real case is $33$.
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HINT:
It is obvious that you need to have the numbers absolute value equal to $1$ (besides one special case but we get to that later). The question is how to set the signs of these $1$s. The product will be negative if you have an odd number of negative signs in it and positive whenever you have an even number of negative signs. Since you are to determine signs for an even number of $1$s if you decide to give an even number of them a negative sign you have one of the following two situations
the $1$ that is NOT in the product has negative sign. In this case there are a $texteven-1=textodd$ number of negative signs distributed along the other five $1$s which result in $-1$ so the condition is fullfilled
the $1$ that in NOT in the product has positive sign. In this case there are a $texteven-0=texteven$ number of negative signs distributed along the five other $1$s which result in $+1$ so the condition is once again fullfilled.
So your question translates to in how many ways can you distribute an even number of negative signs among the six $1$s. This is solved in the easiest way by considering the number of ways to distribute $0,2,4$ and $6$ negative signs.
Finally the one solution not having to deal with $1$s is the one dealing with all zeros.
Hope this helped
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5 Answers
5
active
oldest
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5 Answers
5
active
oldest
votes
active
oldest
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active
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Well, if there are an even number of $1$'s and $-1$'s, then the property holds, so that's $$binom60+ binom62+ binom64+ binom66=32$$
And then there’s the case that they’re all zero. Thus, there are $33$ total cases.
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I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
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– ChakSayantan
Mar 4 at 7:47
109
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You have made no argument as to why this accounts for all cases...
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– Morgan Rogers
Mar 4 at 13:58
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As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
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– Clement C.
Mar 5 at 17:07
2
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@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
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– Zachary Hunter
Mar 6 at 1:08
1
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@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
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– IMil
Mar 6 at 1:13
|
show 1 more comment
$begingroup$
Well, if there are an even number of $1$'s and $-1$'s, then the property holds, so that's $$binom60+ binom62+ binom64+ binom66=32$$
And then there’s the case that they’re all zero. Thus, there are $33$ total cases.
$endgroup$
$begingroup$
I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
$endgroup$
– ChakSayantan
Mar 4 at 7:47
109
$begingroup$
You have made no argument as to why this accounts for all cases...
$endgroup$
– Morgan Rogers
Mar 4 at 13:58
$begingroup$
As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
$endgroup$
– Clement C.
Mar 5 at 17:07
2
$begingroup$
@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
$endgroup$
– Zachary Hunter
Mar 6 at 1:08
1
$begingroup$
@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
$endgroup$
– IMil
Mar 6 at 1:13
|
show 1 more comment
$begingroup$
Well, if there are an even number of $1$'s and $-1$'s, then the property holds, so that's $$binom60+ binom62+ binom64+ binom66=32$$
And then there’s the case that they’re all zero. Thus, there are $33$ total cases.
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Well, if there are an even number of $1$'s and $-1$'s, then the property holds, so that's $$binom60+ binom62+ binom64+ binom66=32$$
And then there’s the case that they’re all zero. Thus, there are $33$ total cases.
edited Mar 5 at 21:42
DAVO
567
567
answered Mar 4 at 7:34
Zachary HunterZachary Hunter
1,065314
1,065314
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I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
$endgroup$
– ChakSayantan
Mar 4 at 7:47
109
$begingroup$
You have made no argument as to why this accounts for all cases...
$endgroup$
– Morgan Rogers
Mar 4 at 13:58
$begingroup$
As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
$endgroup$
– Clement C.
Mar 5 at 17:07
2
$begingroup$
@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
$endgroup$
– Zachary Hunter
Mar 6 at 1:08
1
$begingroup$
@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
$endgroup$
– IMil
Mar 6 at 1:13
|
show 1 more comment
$begingroup$
I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
$endgroup$
– ChakSayantan
Mar 4 at 7:47
109
$begingroup$
You have made no argument as to why this accounts for all cases...
$endgroup$
– Morgan Rogers
Mar 4 at 13:58
$begingroup$
As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
$endgroup$
– Clement C.
Mar 5 at 17:07
2
$begingroup$
@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
$endgroup$
– Zachary Hunter
Mar 6 at 1:08
1
$begingroup$
@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
$endgroup$
– IMil
Mar 6 at 1:13
$begingroup$
I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
$endgroup$
– ChakSayantan
Mar 4 at 7:47
$begingroup$
I was in right direction then. These two last options just confused me. I was thinking there could be much more elements.
$endgroup$
– ChakSayantan
Mar 4 at 7:47
109
109
$begingroup$
You have made no argument as to why this accounts for all cases...
$endgroup$
– Morgan Rogers
Mar 4 at 13:58
$begingroup$
You have made no argument as to why this accounts for all cases...
$endgroup$
– Morgan Rogers
Mar 4 at 13:58
$begingroup$
As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
$endgroup$
– Clement C.
Mar 5 at 17:07
$begingroup$
As Morgan points out, this is fine and all, but this is not a proof. This rules out option A, leaves the others open.
$endgroup$
– Clement C.
Mar 5 at 17:07
2
2
$begingroup$
@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
$endgroup$
– Zachary Hunter
Mar 6 at 1:08
$begingroup$
@IMil, this is true, they were just highlighting that this answer could be more rigorous, which is simply true. the person who answered after me did things much much better in my opinion.
$endgroup$
– Zachary Hunter
Mar 6 at 1:08
1
1
$begingroup$
@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
$endgroup$
– IMil
Mar 6 at 1:13
$begingroup$
@ZacharyHunter sure. I also missed the next obvious step that magnitude has to be either 1 or 0 because otherwise the product will be either more or less than any of the numbers.
$endgroup$
– IMil
Mar 6 at 1:13
|
show 1 more comment
$begingroup$
Here's an argument which extends to $n$ real numbers quite nicely! We can start by noting that
$$x_1 (x_2x_3x_4x_5x_6) = x_1^2$$
And by commutativity, we get $x_i^2 = x_j^2$ which implies that all the magnitudes are equal.
Now if $L$ is the magnitude, we also must have $L = L^5$, and from this we conclude that $L=0$ or $1$.
Now we just have to go through the possibilities. If $L = 0$, then we have all $0$'s
If $L = 1$, then we have all $-1$'s and $1$'s. This configuration works iff the number of $-1$'s is even, as this would imply that
$$fracprod_i=1^6 x_i1 = 1 text and fracprod_i=1^6 x_i-1 = -1$$
Now we can count configurations. There will be
$$sum_i=0^3 binom62i = 2^5$$
possibilities. And finally, we have $1+32 = 33$.
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2
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the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
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– Zachary Hunter
Mar 4 at 17:08
3
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Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
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– Isaac Browne
Mar 4 at 21:03
2
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This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
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– Yakk
Mar 5 at 2:16
2
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
4
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
add a comment |
$begingroup$
Here's an argument which extends to $n$ real numbers quite nicely! We can start by noting that
$$x_1 (x_2x_3x_4x_5x_6) = x_1^2$$
And by commutativity, we get $x_i^2 = x_j^2$ which implies that all the magnitudes are equal.
Now if $L$ is the magnitude, we also must have $L = L^5$, and from this we conclude that $L=0$ or $1$.
Now we just have to go through the possibilities. If $L = 0$, then we have all $0$'s
If $L = 1$, then we have all $-1$'s and $1$'s. This configuration works iff the number of $-1$'s is even, as this would imply that
$$fracprod_i=1^6 x_i1 = 1 text and fracprod_i=1^6 x_i-1 = -1$$
Now we can count configurations. There will be
$$sum_i=0^3 binom62i = 2^5$$
possibilities. And finally, we have $1+32 = 33$.
$endgroup$
2
$begingroup$
the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
$endgroup$
– Zachary Hunter
Mar 4 at 17:08
3
$begingroup$
Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
$endgroup$
– Isaac Browne
Mar 4 at 21:03
2
$begingroup$
This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
$endgroup$
– Yakk
Mar 5 at 2:16
2
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
4
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
add a comment |
$begingroup$
Here's an argument which extends to $n$ real numbers quite nicely! We can start by noting that
$$x_1 (x_2x_3x_4x_5x_6) = x_1^2$$
And by commutativity, we get $x_i^2 = x_j^2$ which implies that all the magnitudes are equal.
Now if $L$ is the magnitude, we also must have $L = L^5$, and from this we conclude that $L=0$ or $1$.
Now we just have to go through the possibilities. If $L = 0$, then we have all $0$'s
If $L = 1$, then we have all $-1$'s and $1$'s. This configuration works iff the number of $-1$'s is even, as this would imply that
$$fracprod_i=1^6 x_i1 = 1 text and fracprod_i=1^6 x_i-1 = -1$$
Now we can count configurations. There will be
$$sum_i=0^3 binom62i = 2^5$$
possibilities. And finally, we have $1+32 = 33$.
$endgroup$
Here's an argument which extends to $n$ real numbers quite nicely! We can start by noting that
$$x_1 (x_2x_3x_4x_5x_6) = x_1^2$$
And by commutativity, we get $x_i^2 = x_j^2$ which implies that all the magnitudes are equal.
Now if $L$ is the magnitude, we also must have $L = L^5$, and from this we conclude that $L=0$ or $1$.
Now we just have to go through the possibilities. If $L = 0$, then we have all $0$'s
If $L = 1$, then we have all $-1$'s and $1$'s. This configuration works iff the number of $-1$'s is even, as this would imply that
$$fracprod_i=1^6 x_i1 = 1 text and fracprod_i=1^6 x_i-1 = -1$$
Now we can count configurations. There will be
$$sum_i=0^3 binom62i = 2^5$$
possibilities. And finally, we have $1+32 = 33$.
edited Mar 5 at 2:36
answered Mar 4 at 7:39
Isaac BrowneIsaac Browne
5,33751334
5,33751334
2
$begingroup$
the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
$endgroup$
– Zachary Hunter
Mar 4 at 17:08
3
$begingroup$
Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
$endgroup$
– Isaac Browne
Mar 4 at 21:03
2
$begingroup$
This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
$endgroup$
– Yakk
Mar 5 at 2:16
2
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
4
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
add a comment |
2
$begingroup$
the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
$endgroup$
– Zachary Hunter
Mar 4 at 17:08
3
$begingroup$
Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
$endgroup$
– Isaac Browne
Mar 4 at 21:03
2
$begingroup$
This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
$endgroup$
– Yakk
Mar 5 at 2:16
2
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
4
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
2
2
$begingroup$
the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
$endgroup$
– Zachary Hunter
Mar 4 at 17:08
$begingroup$
the 2n+1 real number case is pretty much the same, you just add all the cases where there are an even number of “-1”s. so I guess generally it’s $sum_i=0^n binomn2i$
$endgroup$
– Zachary Hunter
Mar 4 at 17:08
3
3
$begingroup$
Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
$endgroup$
– Isaac Browne
Mar 4 at 21:03
$begingroup$
Good point! I'm not sure why I restricted myself to $2n$. Anyways, I'll go and replace $2n$ with $n$ now.
$endgroup$
– Isaac Browne
Mar 4 at 21:03
2
2
$begingroup$
This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
$endgroup$
– Yakk
Mar 5 at 2:16
$begingroup$
This might be clearer if you point out that $x_i^2 = x_j^2$ for all $i$ and $j$ instead of jumping right to "by commutativity". I'm not certain why you called $32$ $2^5$ in your 2nd last line; are you claiming there is an identity easier than $6+10+10+6 = 32$ there?
$endgroup$
– Yakk
Mar 5 at 2:16
2
2
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
$begingroup$
@Yakk As for the former complaint, I'll try to edit that to make it more clear. As for the latter complaint, I wrote $2^5$ because in general we have $$sum_0 leq 2i leq n binomn2i = 2^n-1$$ This identity is why the argument generalizes so nicely!
$endgroup$
– Isaac Browne
Mar 5 at 2:33
4
4
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
$begingroup$
It’s easy to see that the number of subsets of an $n$-set with an even number of elements is always $2^n-1$: just note that the operation of flipping the inclusion of one fixed element is an involution between the odd-sized sets and the even-sized sets.
$endgroup$
– Erick Wong
Mar 5 at 2:43
add a comment |
$begingroup$
Just to make the solution more explicit, the product of all six values is the square of any one of them, so they're all $pm k$ for some $k$, and $k^6=k^2implies kin0,,1$. We get one $k=0$ solution and, because we require a non-negative product, one $k=1$ solution for each choice of six $pm 1$s with an even number of $-1$s. Famously, the numbers of even- and odd-sized subsets of a finite non-empty set are equal, and so the total number of solutions is $1+frac12cdot2^6$, i.e. B)
.
$endgroup$
add a comment |
$begingroup$
Just to make the solution more explicit, the product of all six values is the square of any one of them, so they're all $pm k$ for some $k$, and $k^6=k^2implies kin0,,1$. We get one $k=0$ solution and, because we require a non-negative product, one $k=1$ solution for each choice of six $pm 1$s with an even number of $-1$s. Famously, the numbers of even- and odd-sized subsets of a finite non-empty set are equal, and so the total number of solutions is $1+frac12cdot2^6$, i.e. B)
.
$endgroup$
add a comment |
$begingroup$
Just to make the solution more explicit, the product of all six values is the square of any one of them, so they're all $pm k$ for some $k$, and $k^6=k^2implies kin0,,1$. We get one $k=0$ solution and, because we require a non-negative product, one $k=1$ solution for each choice of six $pm 1$s with an even number of $-1$s. Famously, the numbers of even- and odd-sized subsets of a finite non-empty set are equal, and so the total number of solutions is $1+frac12cdot2^6$, i.e. B)
.
$endgroup$
Just to make the solution more explicit, the product of all six values is the square of any one of them, so they're all $pm k$ for some $k$, and $k^6=k^2implies kin0,,1$. We get one $k=0$ solution and, because we require a non-negative product, one $k=1$ solution for each choice of six $pm 1$s with an even number of $-1$s. Famously, the numbers of even- and odd-sized subsets of a finite non-empty set are equal, and so the total number of solutions is $1+frac12cdot2^6$, i.e. B)
.
answered Mar 4 at 11:49
J.G.J.G.
32.5k23250
32.5k23250
add a comment |
add a comment |
$begingroup$
Even though the question stated real numbers, we can extend this to complex fairly easily.
Note we require
$$ exp( left( E - I) boldsymbolz right ) = exp(boldsymbolz) $$
where $exp(boldsymbolz) = boldsymbolx$ (the vector of numbers to multiply), $E$ is a matrix of ones and $I$ is the identity matrix.
Now,
$$ exp( left( E - 2I) boldsymbolz right ) = exp(2pi i boldsymbolk) $$
for some vector of integers $boldsymbolk$. Equating the powers and using $(E - 2I)^-1 = frac18(E - 4I)$, gives
$$ boldsymbolz = fracK pi i4 - pi i boldsymbolk$$
where $K = sum_i=1^6 (boldsymbolk)_i$ and hence
$$ boldsymbolx = exp left (fracK pi i4 right) cdot exp(pi i boldsymbolk)$$
The second term is clearly a vector of 1s and -1s.
The first term can take the following values:
If the total 1s in the second term are even i.e. (even number of even $k_i$),
- $1$
- $i$
- $-1$
- $-i$
If the total 1s in the second term are odd i.e. (odd number of even $k_i$),
- $frac1sqrt2 + frac1sqrt2i$
- $-frac1sqrt2 + frac1sqrt2i$
- $-(-frac1sqrt2 + frac1sqrt2i)$
- $-(frac1sqrt2 + frac1sqrt2i)$
Finally, consider the subsets of $A = -1, 1^6$, $A_E subset A$ containing an even number of 1s and $A_O subset A$ containing an odd number of 1s.
Clearly $A_E = -A_E$ (multiplying each element in the set by -1) and $A_O = - A_O$, hence the unique combinations of complex numbers is the union of disjoint sets
$$C = A_E cup iA_E cup (frac1sqrt2 + frac1sqrt2i) A_O cup (-frac1sqrt2 + frac1sqrt2i) A_O$$
As $|A_E| = |A_O| = 32$, and the above is a union of disjoint subsets, $|C| = 128$ and including the case where all values are $0$, we have the total complex combinations as $129$.
Note
To answer the question, we know that the only subset containing real values is $A_E$ so the total combinations in the real case is $33$.
$endgroup$
add a comment |
$begingroup$
Even though the question stated real numbers, we can extend this to complex fairly easily.
Note we require
$$ exp( left( E - I) boldsymbolz right ) = exp(boldsymbolz) $$
where $exp(boldsymbolz) = boldsymbolx$ (the vector of numbers to multiply), $E$ is a matrix of ones and $I$ is the identity matrix.
Now,
$$ exp( left( E - 2I) boldsymbolz right ) = exp(2pi i boldsymbolk) $$
for some vector of integers $boldsymbolk$. Equating the powers and using $(E - 2I)^-1 = frac18(E - 4I)$, gives
$$ boldsymbolz = fracK pi i4 - pi i boldsymbolk$$
where $K = sum_i=1^6 (boldsymbolk)_i$ and hence
$$ boldsymbolx = exp left (fracK pi i4 right) cdot exp(pi i boldsymbolk)$$
The second term is clearly a vector of 1s and -1s.
The first term can take the following values:
If the total 1s in the second term are even i.e. (even number of even $k_i$),
- $1$
- $i$
- $-1$
- $-i$
If the total 1s in the second term are odd i.e. (odd number of even $k_i$),
- $frac1sqrt2 + frac1sqrt2i$
- $-frac1sqrt2 + frac1sqrt2i$
- $-(-frac1sqrt2 + frac1sqrt2i)$
- $-(frac1sqrt2 + frac1sqrt2i)$
Finally, consider the subsets of $A = -1, 1^6$, $A_E subset A$ containing an even number of 1s and $A_O subset A$ containing an odd number of 1s.
Clearly $A_E = -A_E$ (multiplying each element in the set by -1) and $A_O = - A_O$, hence the unique combinations of complex numbers is the union of disjoint sets
$$C = A_E cup iA_E cup (frac1sqrt2 + frac1sqrt2i) A_O cup (-frac1sqrt2 + frac1sqrt2i) A_O$$
As $|A_E| = |A_O| = 32$, and the above is a union of disjoint subsets, $|C| = 128$ and including the case where all values are $0$, we have the total complex combinations as $129$.
Note
To answer the question, we know that the only subset containing real values is $A_E$ so the total combinations in the real case is $33$.
$endgroup$
add a comment |
$begingroup$
Even though the question stated real numbers, we can extend this to complex fairly easily.
Note we require
$$ exp( left( E - I) boldsymbolz right ) = exp(boldsymbolz) $$
where $exp(boldsymbolz) = boldsymbolx$ (the vector of numbers to multiply), $E$ is a matrix of ones and $I$ is the identity matrix.
Now,
$$ exp( left( E - 2I) boldsymbolz right ) = exp(2pi i boldsymbolk) $$
for some vector of integers $boldsymbolk$. Equating the powers and using $(E - 2I)^-1 = frac18(E - 4I)$, gives
$$ boldsymbolz = fracK pi i4 - pi i boldsymbolk$$
where $K = sum_i=1^6 (boldsymbolk)_i$ and hence
$$ boldsymbolx = exp left (fracK pi i4 right) cdot exp(pi i boldsymbolk)$$
The second term is clearly a vector of 1s and -1s.
The first term can take the following values:
If the total 1s in the second term are even i.e. (even number of even $k_i$),
- $1$
- $i$
- $-1$
- $-i$
If the total 1s in the second term are odd i.e. (odd number of even $k_i$),
- $frac1sqrt2 + frac1sqrt2i$
- $-frac1sqrt2 + frac1sqrt2i$
- $-(-frac1sqrt2 + frac1sqrt2i)$
- $-(frac1sqrt2 + frac1sqrt2i)$
Finally, consider the subsets of $A = -1, 1^6$, $A_E subset A$ containing an even number of 1s and $A_O subset A$ containing an odd number of 1s.
Clearly $A_E = -A_E$ (multiplying each element in the set by -1) and $A_O = - A_O$, hence the unique combinations of complex numbers is the union of disjoint sets
$$C = A_E cup iA_E cup (frac1sqrt2 + frac1sqrt2i) A_O cup (-frac1sqrt2 + frac1sqrt2i) A_O$$
As $|A_E| = |A_O| = 32$, and the above is a union of disjoint subsets, $|C| = 128$ and including the case where all values are $0$, we have the total complex combinations as $129$.
Note
To answer the question, we know that the only subset containing real values is $A_E$ so the total combinations in the real case is $33$.
$endgroup$
Even though the question stated real numbers, we can extend this to complex fairly easily.
Note we require
$$ exp( left( E - I) boldsymbolz right ) = exp(boldsymbolz) $$
where $exp(boldsymbolz) = boldsymbolx$ (the vector of numbers to multiply), $E$ is a matrix of ones and $I$ is the identity matrix.
Now,
$$ exp( left( E - 2I) boldsymbolz right ) = exp(2pi i boldsymbolk) $$
for some vector of integers $boldsymbolk$. Equating the powers and using $(E - 2I)^-1 = frac18(E - 4I)$, gives
$$ boldsymbolz = fracK pi i4 - pi i boldsymbolk$$
where $K = sum_i=1^6 (boldsymbolk)_i$ and hence
$$ boldsymbolx = exp left (fracK pi i4 right) cdot exp(pi i boldsymbolk)$$
The second term is clearly a vector of 1s and -1s.
The first term can take the following values:
If the total 1s in the second term are even i.e. (even number of even $k_i$),
- $1$
- $i$
- $-1$
- $-i$
If the total 1s in the second term are odd i.e. (odd number of even $k_i$),
- $frac1sqrt2 + frac1sqrt2i$
- $-frac1sqrt2 + frac1sqrt2i$
- $-(-frac1sqrt2 + frac1sqrt2i)$
- $-(frac1sqrt2 + frac1sqrt2i)$
Finally, consider the subsets of $A = -1, 1^6$, $A_E subset A$ containing an even number of 1s and $A_O subset A$ containing an odd number of 1s.
Clearly $A_E = -A_E$ (multiplying each element in the set by -1) and $A_O = - A_O$, hence the unique combinations of complex numbers is the union of disjoint sets
$$C = A_E cup iA_E cup (frac1sqrt2 + frac1sqrt2i) A_O cup (-frac1sqrt2 + frac1sqrt2i) A_O$$
As $|A_E| = |A_O| = 32$, and the above is a union of disjoint subsets, $|C| = 128$ and including the case where all values are $0$, we have the total complex combinations as $129$.
Note
To answer the question, we know that the only subset containing real values is $A_E$ so the total combinations in the real case is $33$.
answered Mar 5 at 11:31
rwolstrwolst
406511
406511
add a comment |
add a comment |
$begingroup$
HINT:
It is obvious that you need to have the numbers absolute value equal to $1$ (besides one special case but we get to that later). The question is how to set the signs of these $1$s. The product will be negative if you have an odd number of negative signs in it and positive whenever you have an even number of negative signs. Since you are to determine signs for an even number of $1$s if you decide to give an even number of them a negative sign you have one of the following two situations
the $1$ that is NOT in the product has negative sign. In this case there are a $texteven-1=textodd$ number of negative signs distributed along the other five $1$s which result in $-1$ so the condition is fullfilled
the $1$ that in NOT in the product has positive sign. In this case there are a $texteven-0=texteven$ number of negative signs distributed along the five other $1$s which result in $+1$ so the condition is once again fullfilled.
So your question translates to in how many ways can you distribute an even number of negative signs among the six $1$s. This is solved in the easiest way by considering the number of ways to distribute $0,2,4$ and $6$ negative signs.
Finally the one solution not having to deal with $1$s is the one dealing with all zeros.
Hope this helped
$endgroup$
add a comment |
$begingroup$
HINT:
It is obvious that you need to have the numbers absolute value equal to $1$ (besides one special case but we get to that later). The question is how to set the signs of these $1$s. The product will be negative if you have an odd number of negative signs in it and positive whenever you have an even number of negative signs. Since you are to determine signs for an even number of $1$s if you decide to give an even number of them a negative sign you have one of the following two situations
the $1$ that is NOT in the product has negative sign. In this case there are a $texteven-1=textodd$ number of negative signs distributed along the other five $1$s which result in $-1$ so the condition is fullfilled
the $1$ that in NOT in the product has positive sign. In this case there are a $texteven-0=texteven$ number of negative signs distributed along the five other $1$s which result in $+1$ so the condition is once again fullfilled.
So your question translates to in how many ways can you distribute an even number of negative signs among the six $1$s. This is solved in the easiest way by considering the number of ways to distribute $0,2,4$ and $6$ negative signs.
Finally the one solution not having to deal with $1$s is the one dealing with all zeros.
Hope this helped
$endgroup$
add a comment |
$begingroup$
HINT:
It is obvious that you need to have the numbers absolute value equal to $1$ (besides one special case but we get to that later). The question is how to set the signs of these $1$s. The product will be negative if you have an odd number of negative signs in it and positive whenever you have an even number of negative signs. Since you are to determine signs for an even number of $1$s if you decide to give an even number of them a negative sign you have one of the following two situations
the $1$ that is NOT in the product has negative sign. In this case there are a $texteven-1=textodd$ number of negative signs distributed along the other five $1$s which result in $-1$ so the condition is fullfilled
the $1$ that in NOT in the product has positive sign. In this case there are a $texteven-0=texteven$ number of negative signs distributed along the five other $1$s which result in $+1$ so the condition is once again fullfilled.
So your question translates to in how many ways can you distribute an even number of negative signs among the six $1$s. This is solved in the easiest way by considering the number of ways to distribute $0,2,4$ and $6$ negative signs.
Finally the one solution not having to deal with $1$s is the one dealing with all zeros.
Hope this helped
$endgroup$
HINT:
It is obvious that you need to have the numbers absolute value equal to $1$ (besides one special case but we get to that later). The question is how to set the signs of these $1$s. The product will be negative if you have an odd number of negative signs in it and positive whenever you have an even number of negative signs. Since you are to determine signs for an even number of $1$s if you decide to give an even number of them a negative sign you have one of the following two situations
the $1$ that is NOT in the product has negative sign. In this case there are a $texteven-1=textodd$ number of negative signs distributed along the other five $1$s which result in $-1$ so the condition is fullfilled
the $1$ that in NOT in the product has positive sign. In this case there are a $texteven-0=texteven$ number of negative signs distributed along the five other $1$s which result in $+1$ so the condition is once again fullfilled.
So your question translates to in how many ways can you distribute an even number of negative signs among the six $1$s. This is solved in the easiest way by considering the number of ways to distribute $0,2,4$ and $6$ negative signs.
Finally the one solution not having to deal with $1$s is the one dealing with all zeros.
Hope this helped
edited Mar 4 at 13:26
answered Mar 4 at 7:19
Vinyl_cape_jawaVinyl_cape_jawa
3,34011433
3,34011433
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9
$begingroup$
They could all be $0$ (unless the image, which I can't access right now, has a rule forbidding that).
$endgroup$
– J.G.
Mar 4 at 7:38
11
$begingroup$
I have typed out your image - pleas do this in the future so users such as @J.G. can view your whole question
$endgroup$
– lioness99a
Mar 4 at 11:13
16
$begingroup$
For extra credit: how many complex numbers? i,i,i,i,i,i,i is one solution. Heck, how many quaternions?.
$endgroup$
– Carl Witthoft
Mar 4 at 13:42
1
$begingroup$
Seems related to the knapsack problem.
$endgroup$
– EJoshuaS
Mar 5 at 16:48
2
$begingroup$
In a similar way to @J.G. they could all be 1.
$endgroup$
– Andrew McOlash
Mar 5 at 19:07