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.)?
777  the real end? I can't seem to get 778. So, I'm going back trying to get every possible "3 five" integer (that's not too big), in hopes of discovering something useful to obtain 778 and of course others.
3 Fives (some already known but missing from the current list):
75 = sqrt(5625) = sqrt(5^5/.5`)
198 = 4.5*44 = (5  .5)*!5
264 = 6*44 = sqrt(5/.5`)!*!5
308 = 7*44 = sqrt(!5+5)*!5
477 = 265*1.8 = (sqrt(5/.5`)!)/.5`
792 = 44*18 = !5/(.5` .5)
1296 = 720*1.8 = sqrt(5/.5`)!!/.5`
and so... 36 = sqrt(1296) = sqrt(sqrt(5/.5`)!!/.5`)
More to come.. I hope.
BTW josh, you have 55 is under "3 Fives." And how did you get 52 and 352 using only three?

Posted by brad
on 20051024 01:25:31 