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.)?

820=600+220=5*(5!+!5) ==>U.Q.

819=U-5/5 etc, covers 813 to 827 , 811, 829

810=u-10

788=u-32

844=u+24

842=u+.5*!5

798=u-22

etc

by now we have to locate as many "3 five" stations as possible

and fiddle around then by +- 7, 10, 22, 24,25, 32, 60 ,120 etc

e.g 1024 us u.q i.e. 4^5

927 is another one! >>>>>>>>> 1854/2=.5*!7

840 is another one .......>>>>>>>>>>7*120

865 =600+265=5*5!+!6

one should not ignore 4 fivers like 884 = 840 + 44 or

1004=1024-120 etc

now building a list becomes more systematic and leaves

only special cases to be attacked

good night now

*Edited on ***October 19, 2005, 11:11 pm**