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jatill
Daedalian Member

 Posted: Tue Oct 05, 2010 8:03 pm    Post subject: 1 1024 is a "powerful number" because 1024 is 1) expressed in simplest form (i.e. no leading 0's, etc), 2) can be represented as x^y, where x and y are integers, and 3) the number of characters in the number 1024 is the same as are the number of characters in x^y (i.e. 2^10 => 4 characters). a) What is the highest powerful number? b) What is the lowest powerful number? c) What is the powerful number closest to 0?
Zag
Tired of his old title

 Posted: Tue Oct 05, 2010 8:19 pm    Post subject: 2 For the lowest, how about: 4^2 or 2^4 = 16 Of course, you probably should have said the lowest in absolute value. There are negative powerful numbers, I'm sure. (-27, for instance = (-3)^3, which, I think, is the largest negative powerful number.)
bonanova
Daedalian Member

 Posted: Tue Oct 05, 2010 11:52 pm    Post subject: 3 Zag, does your LPN candidate obey (3)? Maybe I don't understand (3)._________________ Vidi, vici, veni.
Trojan Horse
Daedalian Member

 Posted: Wed Oct 06, 2010 12:01 am    Post subject: 4 For the one closest to 0: .000001=1000^-2 (with 7 characters in each) is the best I've been able to come up with so far. (Assuming that's not cheating.)
Zag
Tired of his old title

 Posted: Wed Oct 06, 2010 1:15 am    Post subject: 5 Oh. I didn't read the rule carefully enough to realize we were supposed to count the caret. Of course, when you write it in normal math style, there isn't one.
Jake2*
Guest

 Posted: Wed Oct 06, 2010 11:20 am    Post subject: 6 For the largest, I get 99856 (316^2).
jatill
Daedalian Member

Posted: Wed Oct 06, 2010 12:04 pm    Post subject: 7

 Zag wrote: For the lowest, how about: 4^2 or 2^4 = 16 Of course, you probably should have said the lowest in absolute value. There are negative powerful numbers, I'm sure. (-27, for instance = (-3)^3, which, I think, is the largest negative powerful number.)

By lowest I did mean most negative, otherwise it would have been the same as (c).

Edit: (a) and (b) are both unsolved. I don't actually know the answer for (c), but don't forget that .25 and 1/4 are valid numbers
Trojan Horse
Daedalian Member

Posted: Wed Oct 06, 2010 4:38 pm    Post subject: 8

 jatill wrote: Edit: (a) and (b) are both unsolved.

Did you miss Jake2's answer, jatill? I'm going to defend him, and say that he got the best possible answer for (a). (Assuming that, in the expression x^y, both x and y must be written in "standard form", with no leading zeros.)

I think I could write out a full proof that Jake2 got the best possible answer. (No time to do it right now, though.)
jatill
Daedalian Member

Posted: Wed Oct 06, 2010 5:01 pm    Post subject: 9

Trojan Horse wrote:
 jatill wrote: Edit: (a) and (b) are both unsolved.

Did you miss Jake2's answer, jatill?

Nope, but I'm pretty sure there's a better one.
Death Mage
Raving Lunatic

Posted: Wed Oct 06, 2010 5:23 pm    Post subject: 10

 Quote: For the one closest to 0: .000001=1000^-2 (with 7 characters in each) is the best I've been able to come up with so far. (Assuming that's not cheating.)

Do decimal commas count?
-.00000001 = -10,000^-2
_________________
* These senseless ramblings brought to you by Insanity™. If you just can't figure the dang thing out, it must be Insanity™.
jatill
Daedalian Member

Posted: Wed Oct 06, 2010 5:36 pm    Post subject: 11

Death Mage wrote:
 Quote: For the one closest to 0: .000001=1000^-2 (with 7 characters in each) is the best I've been able to come up with so far. (Assuming that's not cheating.)

Do decimal commas count?
-.00000001 = -10,000^-2

Commas don't count. The only characters that are legal would be 0-9, -, ., and / (for fractions).
Trojan Horse
Daedalian Member

 Posted: Wed Oct 06, 2010 8:20 pm    Post subject: 12 Oh, duh... [998001=(-999)^2] I'm assuming, based on the last comment, that parentheses don't count. If they did count, we could do this: [9999800001=(-99999)^2]
jatill
Daedalian Member

Posted: Thu Oct 07, 2010 12:07 pm    Post subject: 13

 Trojan Horse wrote: Oh, duh... [998001=(-999)^2] I'm assuming, based on the last comment, that parentheses don't count. If they did count, we could do this: [9999800001=(-99999)^2]

You're on the right track, but you can do better. And you are right in assuming that parens are illegal.
Trojan Horse
Daedalian Member

 Posted: Thu Oct 07, 2010 8:17 pm    Post subject: 14 *sigh* Probably still not the best, but [9998244=-3162^2] works. I await your reply that there is still a better answer out there.
jatill
Daedalian Member

 Posted: Fri Oct 08, 2010 10:20 pm    Post subject: 15 You finally nailed part (a)
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