Justin
2006-02-03 21:50:04 |
Five Fives
Anybody remember that old problem? I just looked at it and noticed that primorial (#) wasn't used. Using it, I solved for the previously "impossible" number of 779. Maybe it's time to take a look at the problem again. Any takers? |
brianjn
2006-02-04 02:17:49 |
Re: Five Fives
Primorial??
I had a sense of some a prime concept but could not come to grips with it. I wonder how many others who were actually following the "Five Fives" theme actually knew about the concept.
I found this link:
<url>http://mathworld.wolfram.com/Primorial.html <url>
|
brianjn
2006-02-04 02:18:59 |
Re: Five Fives
Forget the tags. |
Justin
2006-02-04 09:23:59 |
Re: Five Fives
I knew about primorial before I heard of subfactorial, and the other strange functions, so I think it should be completely valid. Do other people agree? |
Mindrod
2006-02-04 14:21:21 |
Re: Five Fives
I solved for 806 using primorial. It looks like a valid function to me.
<url>http://primes.utm.edu/glossary/page.php?sort=Primorial</url> |
Mindrod
2006-02-04 14:41:40 |
Re: Five Fives
Here's another way to solve 806, introducing another new function "||" to the mix:
806 = [(5 + 5# + 5)||(5#)]/5 |
Mindrod
2006-02-04 14:57:21 |
Re: Five Fives
Perhaps I should have used parentheses instead of brackets in my last post. Brackets might be used for something else.
"||" is the symbol used to denote concatenation, although the symbol "<>" is used in "Mathematica" (Wolfram Research, Champaign, IL).
The concatenation of two numbers is the number formed by concatenating their numerals. The concatenation of 40 and 30, for example, is 4030. The value of the result depends on the numeric base. The formula for the concatenation of numbers n and m in base 10 is
m||n = (m x 10^([log n] + 1)) + n
where the brackets are used to denote the Floor Function. |
Justin
2006-02-04 15:51:47 |
Re: Five Fives
Does anyone think that this problem deserves another shot of attention, or is it dead? Maybe going beyond 820, seeing as how we have 779, 778 is easier, and 806 is solved. I might even spend my study hall doing this. |
Mindrod
2006-02-04 16:57:58 |
Re: Five Fives
Certainly. We might alos suggest improvements to numbers already posted.
For example, instead of 22 = 55/(5 x .5), which uses 4 fives, we might suggest 22= !5 x .5, which only uses 2 fives.
http://www.geocities.com/josh70679/FiveFives.html |
Gamer
2006-02-04 18:12:42 |
Re: Five Fives
It depends on what your definition of improvement is. I do not initially know what !5 is, and so it seems more complex to have a lesser known function than it does to have 4 5s. |
Mindrod
2006-02-04 19:09:28 |
Re: Five Fives
My definition of improvement is a solution that uses fewer 5s so that it might be used as a building block for finding solutions for additional numbers. Also, solutions with fewer symbols might be considered better than solutions with more symbols. For example, 1 = 5/5 is better than 1 = 5!/5! |
Justin
2006-02-04 23:09:39 |
Re: Five Fives
So improvements like, 30=5# rather than √(5/.5)! * 5, and 31 = 5# + 5/5, or just use the originals? After all, some may not know about primorial, just like I didn't know about subfactorial. |
Justin
2006-02-04 23:32:16 |
Re: Five Fives
That box should be the symbol for a repeating digit. So that fraction is 5/.5555555555........ = 9. |
brianjn
2006-02-05 01:42:15 |
Re: Five Fives
The guideline has been set as "Five Fives"; we cannot go outside of that.
Certainly we create a new thread, say "Less that 5 Fives for Five Fives Solutions".
Leave the 'purity' of "Five Fives" but setup a 'parallel' theme. |
Mindrod
2006-02-05 02:24:04 |
Re: Five Fives
Many of the solutions in the "Five Fives" thread use less than five fives. Five, however, is the maximum allowable number of "5"s that may be used. Our objective in finding solutions that use fewer fives is not to replace existing solutions with "better" solutions -- any solution using no more than 5 fives is a good one. Our objective is to free up fives, so that the solved number can be used as a building block for finding solutions to other numbers, using the remaining fives. For this purpose, the more fives left over the better. |
Justin
2006-02-05 09:30:44 |
Re: Five Fives
I was just putting in the actual 5 that was repeated, because the repetition symbol didn't work in the previous post. In other words, the tilde wasn't working right, so I put the multiple fives in. Sorry about the confusion. And, I know that I should be replacing things like the first solution of 30 = sqrt(9)!*5 where 9 is the fraction from above, with the simplified version 30 = 5#. That cuts it down from 3 fives, to 1 five, making it a very "useful quantity". |
Justin
2006-02-05 11:22:17 |
Re: Five Fives
After you mentioned improvement, I took a look at official list of solutions at Josh's site. I noticed that 6!=720 is used in multiple solutions. Without primorial, we have to use 6!=(sqrt(5/.5`)!)!. But with primorial, 6! = (5#/5)!. We eliminate the radical, the repeated digit, and one factorial. Plus, (5#/5)! can be typed with standard characters, and won't vary. That is we won't see ` sometimes, and tilde others. Makes for a more standardized layout. |
Mindrod
2006-02-05 23:38:47 |
Re: Five Fives
Another useful function is the superfactorial (Mathworld). 5$=34,560 doesn't seem useful at first glance, but 5$/5! = 288 is a useful quantity. |
Jer
2006-02-06 12:54:45 |
Re: Five Fives
You five 5's hunters should also be aware of double-factorial
n!! = n(n-2)(n-4)...*3*1 if n is odd
n!! = n(n-2)(n-4)...*2 if n is even
so 5!!=15
That might help. |
Justin
2006-02-08 10:20:22 |
Re: Five Fives (near 1000)
I've solved up to 978! Only 21 numbers left under 1000! We might break it soon. |
Justin
2006-02-19 07:18:30 |
Re: Five Fives
Congratulations to Dej Mar for finding solutions for 979 to 1000. Now, to see if we can go further. |
