In this set of alphametic relationships, each of the letters in bold denotes a different digit from 0 to 9. None of the numbers can contain any leading zero.

- 35*(
**PLUMS**) < 2*(**MELONS**) < 3*(**APPLES**), and:
- 16*(
**APPLES**) < 99*(**LEMONS**) < 210*(**PLUMS**)

What is the number that is represented by

**BANANA**?

I used a mixture of both mind and machine (mostly machine) to conclude that (0123456789) = (ELONUPABSM), thus **(BANANA) = 763,636**.

I first noted that since 35*(PLUMS) < 3*(APPLES), then 210*(PLUMS) < 18*(APPLES). Therefore 16*(APPLES) < 99*(LEMONS) < 210*(PLUMS) < 18*(APPLES). That meant 99 LEMONS were squeezed between 16 and 18 APPLES, leading me to think that 99*L must be close to 17*A. Then L could only be 1, with A likely 6 or possibly 5. I ran a program with the constraints L=1 and A=6 to find out that P=5, E=0, and M=7, 8, or 9; trying A=5 gave no results. I then ran a similar program with these new constraints to find that M=9, O=2, N=3, S=8, and U=4, and therefore the unused B=7.

And that's how I spent my entire Saturday, and I can't think of a better way to do it!

*Edited on ***October 18, 2009, 1:44 am**