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Topic: Speaking of cubes... 


What's the smallest number than can be expressed as the sum of two cubes...in two different ways?! 


.. don't suppose you are going to accept (1^3)+(2^3)=9 and (2^3)+(1^3)=9 


[quote] On 20060317 04:38, magicjohn2278 wrote: .. don't suppose you are going to accept (1^3)+(2^3)=9 and (2^3)+(1^3)=9 [/quote] ...and if not, then 1729... 


Zero, in an infinite number of ways. (2^3) + (2^3) = 0 (3^3) + (3^3) = 0 


There is no smallest number with that reasoning, Daegs. You can probably come up with a few examples where the result is smaller than Zero. /Tomas 


1^3 + 12^3 and 9^3 + 10^3 = 1729 (apparently) What's the smallest number than can be expressed as the sum of two cubes...in three different ways?! 


[quote] On 20060317 08:04, Steve Martin wrote: 1^3 + 12^3 and 9^3 + 10^3 = 1729 (apparently) What's the smallest number than can be expressed as the sum of two cubes...in three different ways?! [/quote] You've GOT to be kidding! 


[quote] On 20060317 08:04, Steve Martin wrote: What's the smallest number than can be expressed as the sum of two cubes...in three different ways?! [/quote] 87539319 


... which, coincidentally, is my phone number. 


I wouldnt really say negative numbers are "smaller" than zero, quiet the opposite. Zero is the smallest number, at least imho. 


[quote] On 20060317 23:26, Daegs wrote: Zero is the smallest number, at least imho. [/quote] Expect some trouble when it comes to inequalities, logic and programming. ;) /Tomas 


Well certainly 4 < 0 but which is "smaller"? smaller /larger refers to reality and in reality there is no zero or negative numbers.... which is why I posted what I did. The way I see it (when it comes to smaller/larger), is that negative numbers have a greater magnitude in the opposite direction of positives. Viewing magnitude, I see zero as "smaller" than 1000 for example. ANYWAY, if you look at it that way that less than = smaller, then the answer is negative infinity as you can keep giving it larger and larger negatives to be cubed and added.. All in all a very poorly worded puzzle imho.... 


The words "smallest number" was probably a hint that he was talking about maths. Zero and negative you of course find in reality. Your teacher probably explained zero as "none" and negative as "owing". Nevertheless the problem poser spoke of "number" and not fruits or animals. I agree that the wording of the puzzle could have been stricter, but it's easy to reason that "There is no smallest number." is such a boring answer that he must have meant something else. Since I'm not very good with Diophantine problems, did you guys try different solutions or was there some other way of solving this? /Tomas 


I had 1729 in mind, but interesting theoretical followup you guys had while I was out of town. ;) 