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Five Fives (Posted on 2005-09-18) Difficulty: 4 of 5
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.)?

See The Solution Submitted by Josh70679    
Rating: 4.4737 (19 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
end of the line | Comment 304 of 392 |
(In reply to my final(for today) words by Ady TZIDON)

As much as I hate to admit it, I think the problem is finished.  My original goal for this problem was to see if getting to 555 was remotely possible (i didn't think it was). I'm so impressed that we managed it, I don't think going further will make it any more impressive.  Thanks and congratulations to everyone who helped to make this problem such a smashing success!

Of course, the problem is still open as to the highest consecutive integer that can be expressed.  If anyone wants to keep working, I'm more than happy to continue to update the list.

Edited on October 15, 2005, 7:07 am
  Posted by Josh70679 on 2005-10-15 07:03:08

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