A mathematician who was exceedingly fond of the number five set to work trying to express as many consecutive integers using no numerals besides '5', and only up to five of them. She allowed herself to use any standard mathematical notation she knew, as long as it didn't contain any numerals. For example, she could use the symbol for 'square root', but not 'cube root' (because it contains a '3'). She determined that the highest consecutive integer she could express this way was 36. Her last few calculations were as follows:

• 31 = 5*5 + 5 + (5/5)
• 32 = 55*.5 + 5 - .5
• 33 = (55 + 5) * .55
• 34 = 5!/5 + 5/.5
• 35 = (5 + (5+5)/5) * 5
• 36 = 5*5 + 55/5
• 37 = ?

Was she correct in thinking 36 was the highest consecutive integer she could express this way? Can you express 37 using only up to five 5's?

Note: The intention here is to find an exact expression, so rounding expressions like [] "greatest integer" are not allowed.
Note: Can you do it without using letters of any kind (x, log, lim, sum, etc.)?

Fellow Perplexus-ites,

I am in awe of what you have done here.  I decided to join in late and try the challenge of 778, but figured I needed to prove the first 777 numbers to feel I was legit.

I was able to do 775 of the first 777 numbers with the aid of the subfactorial function.  I ran into a couple of problems with two numbers in the mid 400's. Refering to Josh's list, I reconciled that those two were do-able.

I am 99% convinced that 778, 779 and 806 are not do-able with the functions provided.  But it is the remaining 1% that makes me ponder this problem further. . .

Posted by Leming on 2005-12-07 13:04:37