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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.)
sound off about % | Comment 240 of 391 |

I'm conflicted about what constitutes correct usage of '%'.  Here are a few rhetorical questions i've been kicking around:

  • 1) Must % be used as a scalar?
  • I have no problem using % as a scalar, e.g. "6 = 5!*5%"
  • I'm not sure using it as a constant expression is allowed, though, e.g. "101 = (5 + 5%) * (5/(.5*.5))."

  • 2) Is % an operation? 
  • i.e. is adding % to an expression equivalent to adding "/100"? 
  • In all cases, an explicit number followed by % is ok, for example "5%" or "555.5%".
  • It doesn't necessarily follow, though, that any expression followed by % is allowed, e.g. "(5!)%"

Let me know what you think.  Do you have a strong opinion about this?  Do you know of a reference that would clear this up?  If we can't resolve this another way, we could always vote on what to allow for this problem. 


  Posted by Josh70679 on 2005-10-10 16:18:49
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