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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.)
re: to Josh | Comment 137 of 392 |
(In reply to to Josh by pcbouhid)

sure thing, though i haven't been actively registering them all.  i'd be glad to compile a list once we decide to stop (if that ever happens ;)). 

One thing I ask: earlier in the posts, some people claimed to have solutions but didn't write them out.  To those people, please post them!  it will make my life a lot easier.  Don't worry about 1 thru 36, i have those. 

Anyone else who wants our final list, reply to this thread and let me know how to get it to you. 

  Posted by Josh70679 on 2005-09-21 20:53:38
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