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July 24, 2025, 01:27:44 pm

Author Topic: cute puzzle  (Read 2287 times)  Share 

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zzdfa

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cute puzzle
« on: July 03, 2009, 12:02:28 am »
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hi guyz

i thought the solution to this was quite elegant:

I have 4 (not necessarily integer) numbers, say m,n,o,p.

Now obviously there are 6 ways to multiply them:

mn no op
mo np
mp

I tell you that of these 6 products, 5 of them are:

2,3,4,5,6. What's the 6th one equal to?

kamil9876

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Re: cute puzzle
« Reply #1 on: July 03, 2009, 12:05:05 am »
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I often show public displays of affection towards prime factors on this forum.
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

TrueTears

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Re: cute puzzle
« Reply #2 on: July 03, 2009, 12:05:44 am »
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I often show public displays of affection towards prime factors on this forum.
You and your prime.
PhD @ MIT (Economics).

Interested in asset pricing, econometrics, and social choice theory.

kamil9876

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Re: cute puzzle
« Reply #3 on: July 03, 2009, 01:33:49 am »
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we can assume, WLOG, that is what we do not have. Notice that:


But the only products satisfying such a ratio are 2 and 4;3 and 6. Hence we can assign, WLOG, that and so this leaves us with .






similairly:





Discarding the negative because it yields a contradiction since if op is negative then one of the numbers must be negative. Assume p is negative, this implies that n is negative since np is positive. Now if n is negative then that implies that m is negative since mn is positive. But that implies that mo must be negative(since we assumed o is positive), hence implying that one of the 5 products you provided us with is negative which yields a contradiction.
Same argument can be applied if p is positive hence the law of the excluded middle implies that cannot be negative.
« Last Edit: July 03, 2009, 05:46:47 pm by kamil9876 »
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

zzdfa

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Re: cute puzzle
« Reply #4 on: July 03, 2009, 11:19:56 am »
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op cant be 6/5, check your working

kamil9876

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Re: cute puzzle
« Reply #5 on: July 03, 2009, 05:47:27 pm »
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yep sorry, edited now, forgot the factor of 4 which eventually doubles my answer after square rooting :)
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

zzdfa

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Re: cute puzzle
« Reply #6 on: July 03, 2009, 06:54:18 pm »
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nice
well, here's an easier way:

note that:






you pretty much had it in the first 2 lines  :P
« Last Edit: July 03, 2009, 07:02:13 pm by zzdfa »

kamil9876

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Re: cute puzzle
« Reply #7 on: July 03, 2009, 10:34:00 pm »
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yeah i guess i just included a lot of words and explanations, giving an impression that it had many computations but it didnt. Still i reckon yours has a less computations so it's probably better, but it's harder to see everything fit in like that imo. Nice stuff.
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."