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**Don't try this puzzle**

It looks childish, but this puzzle is sadistically difficult. Saying that 95% of people can't solve this is like saying 95% of people can't jump over a skyscraper.

Here is the simplest solution:

apple = 154476802108746166441951315019919837485664325669565431700026634898253202035277999

banana =

36875131794129999827197811565225474825492979968971970996283137471637224634055579

pineapple =

4373612677928697257861252602371390152816537558161613618621437993378423467772036

You need a serious course on number theory to learn how to solve this. So it's easier than jumping over a skyscraper: you can learn to do it. But without some education, it's pretty much impossible.

The trick is to transform the equation into an

**elliptic curve**. An elliptic curve is a kind of curve whose points form a group. That means if you find one point on the curve, you can find more. So if you can find

*one*solution of this puzzle, you can find more.

Umm, but then you still need to find a solution! Luckily there's a small solution where the variables are integers that

*aren't*positive:

apple = 4

banana = -1

pineapple = 11

From this you can turn the crank and get more solutions, but they get bigger and bigger, and the first one where all three variables are positive is the one I showed you.

I got all this from a wonderful Quora post by Alon Amit:

https://www.quora.com/How-do-you-find-the-integer-solutions-to-frac-x-y+z-+-frac-y-z+x-+-frac-z-x+y-4/answer/Alon-Amit

but I heard about

*that*from +David Eppstein, here on G+. So: add David Eppstein to your list of cool people you follow on G+!

The post by Alon Amit is worth reading, because he leads you through the number theory without getting too technical (leaving out lots of juicy details that you'd get in a course on elliptic curves), and he gives some examples of similar problems that are

*much harder*- if you don't know the trick.

#bigness

View 39 previous comments

- +John Baez see the page titled "A Divisor Relation on Abelian Varieties" in this pdf: msri.org - www.msri.org/attachments/workshops/301/HtSurveyMSRIJan06.pdf

It's stated for abelian varieties, which of course includes elliptic curves. So the height of mP grows quadratically for fixed P and m=1,2,3,...

We can take the divisor D to be the point at infinity=identity for the elliptic curve.6w - +David Roberts - nice! Unfortunately that quadratic growth holds asymptotically, so we only know that
*eventually*the height of nX (using our earlier notation) will become big and nX must be a more complicated solution than 9X.

If we had a more precise estimate, we'd get an explicit bound on "eventually".

That would reduce the proof that 9X is the simplest solution to a finite search.6w - +John Baez if one uses "canonical" height, rather than naive height, one gets quadratic growth on the nose, but canonical height is defined as a limit as well, so the connection to numbers of digits or mere size of solutions is not clear.6w
- +David Roberts - so, for now we may have to live with the possibility that some kid will come up with a simpler answer to this puzzle. :-)6w
- Chris McLelland writes:
~~----------------------------------------------~~

The physicist in me proposes the following solution:

Strawberry: 35

Banana: 132

Pineapple: 627

...because 4.000000095 is approximately 4.0.

Or even better:

Strawberry: 688

Banana: 1599

Pineapple: 8600

Result: 4.00000000018

:-)6w - assume b=1, c=1 and find solution for f(a)=4 ?

this gives us a quadratic equation a*a-7a-4=0

then a = (7+ sqrt(65))/2 lol6w - 6w

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