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 2012-04-03, 19:03 #1 bcp19     Oct 2011 2A716 Posts Lotto Primes With the recent lotto fever, I got to thinking about a puzzle. Given 6 lotto numbers, 5 ranged 1-56 and 1 ranged 1-46(010203040501 to 525354555646), there are 720 combinations that each 6 number result can be arranged. What set of numbers gives the most % of prime numbers? While I am not an up-to-date programmer, I can use the excel macro 'program' fairly well. Using this I was able to put together a program to test combinations, but due to it being so slow, it would take 46 years to finish. My best find thus far has been 153 primes out of the 720 combinations, can anyone beat that?
2012-04-03, 19:15   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

2·5·839 Posts

Quote:
 Originally Posted by bcp19 With the recent lotto fever, I got to thinking about a puzzle. Given 6 lotto numbers, 5 ranged 1-56 and 1 ranged 1-46(010203040501 to 525354555646), there are 720 combinations that each 6 number result can be arranged. What set of numbers gives the most % of prime numbers? While I am not an up-to-date programmer, I can use the excel macro 'program' fairly well. Using this I was able to put together a program to test combinations, but due to it being so slow, it would take 46 years to finish. My best find thus far has been 153 primes out of the 720 combinations, can anyone beat that?
16 primes under 56 ( 14 of which are under 46) really the only complexity is the less than 46 part the other part is $\frac{16!}{5! \times 11!}$ the maximum below 46 left for any one of these is 11. then it's as simple as answer / 720 for all prime ( 100%) combinations.

Last fiddled with by science_man_88 on 2012-04-03 at 19:25

2012-04-03, 19:37   #3
bcp19

Oct 2011

7·97 Posts

Quote:
 Originally Posted by science_man_88 16 primes under 56 ( 14 of which are under 46) really the only complexity is the less than 46 part the other part is $\frac{16!}{5! \times 11!}$ the maximum below 46 left for any one of these is 11. then it's as simple as answer / 720 for all prime combinations.
I don't think you understand the problem posed. Break it down to 3 numbers, and take numbers 1,3,6. Since the problem deals with single and double digit numbers, this is represented as 10306, 10603, 30106, 30601, 60103 and 60301 of which only 60103 is prime.

 2012-04-03, 21:23 #4 axn     Jun 2003 23·673 Posts These are the 164 primes for 3,7,17,21,43,49. Not sure your problem specification allows for 49 to come in the last place. Code: 30717432149 30717434921 30721431749 30721491743 31707214349 31707492143 31707494321 31721074349 31721074943 31743214907 31749214307 31749430721 32143174907 32149174307 32149430717 34317072149 34317210749 34317214907 34349170721 34349172107 34349210717 34349211707 34907211743 34907431721 34917210743 34921170743 34943072117 34943211707 70317434921 70317492143 70321174943 70321431749 70343491721 70349431721 70349432117 71721430349 71743032149 71743034921 71743214903 71749034321 72103491743 72117034943 72143034917 72143170349 72143490317 72149031743 72149170343 72149431703 74317492103 74321490317 74321491703 74349032117 74903172143 74903431721 74903432117 74917432103 74943031721 74943170321 74943210317 170307492143 170343210749 170343214907 170703214349 170721034349 170721494303 170743492103 170749034321 170749432103 172103494307 172149030743 174303074921 174303492107 174307034921 174307490321 174307492103 174349032107 174349072103 174903072143 174903210743 174907034321 174921070343 174943032107 210307174943 210307494317 210343174907 210343490717 210703174943 210703431749 210749031743 210749431703 211703074943 211703490743 211707034943 211707430349 211707434903 211743074903 211749034307 211749430307 214307031749 214307034917 214307174903 214317030749 214349170307 214903074317 214903174307 214903431707 214907031743 214943031707 214943071703 214943170307 214943170703 430307214917 430317490721 430321490717 430321491707 430349170721 430349210717 430703172149 430717210349 430721034917 430749032117 430749210317 431707034921 431721070349 431749030721 431749070321 432103071749 432103490717 432107031749 432107490317 432149031707 432149170307 432149170703 434903072117 434903170721 434907172103 434907210317 434917070321 434921071703 490307431721 490317210743 490317214307 490321174307 490343071721 490343170721 490343172107 490343210717 490703214317 490703431721 490717214303 490721031743 490721431703 490743031721 491707210343 491743030721 491743070321 492103430717 492107034317 492107431703 492117070343 494303172107 494303210717 494307031721 494317210307
2012-04-03, 21:33   #5
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

2·5·839 Posts

Quote:
 Originally Posted by axn These are the 164 primes for 3,7,17,21,43,49. Not sure your problem specification allows for 49 to come in the last place. Code: 30717432149 30717434921 30721431749 30721491743 31707214349 31707492143 31707494321 31721074349 31721074943 31743214907 31749214307 31749430721 32143174907 32149174307 32149430717 34317072149 34317210749 34317214907 34349170721 34349172107 34349210717 34349211707 34907211743 34907431721 34917210743 34921170743 34943072117 34943211707 70317434921 70317492143 70321174943 70321431749 70343491721 70349431721 70349432117 71721430349 71743032149 71743034921 71743214903 71749034321 72103491743 72117034943 72143034917 72143170349 72143490317 72149031743 72149170343 72149431703 74317492103 74321490317 74321491703 74349032117 74903172143 74903431721 74903432117 74917432103 74943031721 74943170321 74943210317 170307492143 170343210749 170343214907 170703214349 170721034349 170721494303 170743492103 170749034321 170749432103 172103494307 172149030743 174303074921 174303492107 174307034921 174307490321 174307492103 174349032107 174349072103 174903072143 174903210743 174907034321 174921070343 174943032107 210307174943 210307494317 210343174907 210343490717 210703174943 210703431749 210749031743 210749431703 211703074943 211703490743 211707034943 211707430349 211707434903 211743074903 211749034307 211749430307 214307031749 214307034917 214307174903 214317030749 214349170307 214903074317 214903174307 214903431707 214907031743 214943031707 214943071703 214943170307 214943170703 430307214917 430317490721 430321490717 430321491707 430349170721 430349210717 430703172149 430717210349 430721034917 430749032117 430749210317 431707034921 431721070349 431749030721 431749070321 432103071749 432103490717 432107031749 432107490317 432149031707 432149170307 432149170703 434903072117 434903170721 434907172103 434907210317 434917070321 434921071703 490307431721 490317210743 490317214307 490321174307 490343071721 490343170721 490343172107 490343210717 490703214317 490703431721 490717214303 490721031743 490721431703 490743031721 491707210343 491743030721 491743070321 492103430717 492107034317 492107431703 492117070343 494303172107 494303210717 494307031721 494317210307
as long as one of them is under 46 it's okay it seems.

 2012-04-03, 21:42 #6 axn     Jun 2003 23·673 Posts A few results above 150. Code: 152 [3,11,13,17,27,29] 155 [1,3,7,9,17,31] 152 [3,11,19,21,27,31] 155 [9,17,21,23,27,31] 152 [1,3,7,9,29,31] 159 [1,7,11,21,29,31] 159 [3,7,9,21,27,33] 156 [7,19,21,27,29,33] 150 [9,19,21,23,31,33] 158 [1,3,11,23,29,37] 153 [9,11,19,23,29,37] 150 [7,9,17,27,31,37] 156 [1,11,17,27,31,37] 151 [1,9,23,29,31,37] 152 [1,3,21,29,33,37] 151 [11,13,19,23,29,39] 157 [1,13,23,27,33,39] 157 [3,17,29,31,33,39] 151 [7,11,13,21,37,39] 150 [1,13,23,27,37,39] 165 [3,7,11,31,37,39] 151 [9,11,31,33,37,39] 152 [3,7,9,13,21,41] 154 [9,11,13,19,23,41] 151 [9,13,21,23,27,41] 150 [3,11,17,27,31,41] 151 [1,13,19,27,33,41] 151 [7,9,11,29,33,41] 159 [3,13,27,31,33,41] 150 [1,3,21,33,37,41] 150 [19,27,31,33,37,41] 150 [1,3,9,23,39,41] 165 [3,11,13,27,39,41] 150 [7,19,27,33,39,41] 151 [1,3,13,17,27,43] 151 [7,11,17,19,27,43] 152 [1,3,11,17,29,43] 150 [3,11,17,19,31,43] 158 [1,3,11,29,31,43] 150 [1,7,13,21,33,43] 163 [7,17,19,29,33,43] 155 [1,3,7,13,37,43] 150 [3,7,17,29,37,43] 159 [1,13,19,29,37,43] 151 [9,13,21,29,37,43] 154 [1,21,27,31,37,43] 154 [7,17,29,31,37,43] 150 [1,11,13,17,39,43] 155 [3,7,17,19,39,43] 159 [3,19,21,23,39,43] 152 [1,9,23,33,39,43] 153 [13,27,29,33,39,43] 158 [1,9,13,27,41,43] 151 [1,3,7,29,41,43] 166 [9,13,23,37,41,43] 152 [1,9,29,37,41,43] 151 [1,11,13,19,21,47] 152 [11,19,21,23,27,47] 156 [1,7,11,19,31,47] 159 [1,3,17,19,31,47] 151 [3,17,19,23,33,47] 154 [1,3,21,29,33,47] 156 [11,17,27,29,33,47] 151 [3,17,27,31,33,47] 157 [9,17,19,23,37,47] 162 [3,17,23,31,37,47] 152 [3,9,17,21,39,47] 162 [1,17,27,33,39,47] 153 [3,21,29,33,39,47] 150 [3,13,17,27,41,47] 157 [1,13,27,31,41,47] 150 [11,17,29,33,41,47] 152 [1,13,31,39,41,47] 150 [1,31,37,39,41,47] 151 [7,13,21,33,43,47] 154 [1,19,27,33,43,47] 151 [7,21,27,33,43,47] 150 [1,7,29,33,43,47] 153 [1,11,33,37,43,47] 153 [1,3,7,13,21,49] 166 [1,3,7,21,23,49] 153 [7,11,13,21,23,49] 159 [3,7,13,23,27,49] 157 [1,13,23,27,29,49] 151 [3,7,21,29,31,49] 156 [3,9,19,21,33,49] 150 [3,7,19,29,33,49] 150 [1,17,21,31,33,49] 157 [1,9,11,33,37,49] 160 [3,17,21,33,37,49] 151 [1,9,21,27,39,49] 163 [1,21,29,33,39,49] 154 [1,3,19,21,41,49] 154 [13,19,21,29,41,49] 151 [3,7,19,33,41,49] 166 [1,3,17,37,41,49] 164 [3,7,17,21,43,49] 155 [1,17,21,29,43,49] 155 [11,21,29,31,43,49] 150 [1,11,23,37,43,49] 169 [1,3,31,37,43,49] 153 [1,13,37,41,43,49] 150 [1,7,13,19,47,49] 152 [9,11,19,37,47,49] 152 [7,13,17,21,27,51] 150 [3,17,19,23,27,51] 160 [1,9,11,23,29,51] 156 [3,7,9,23,31,51] 151 [3,17,21,23,31,51] 151 [7,9,17,23,33,51] 158 [1,3,19,27,33,51] 153 [9,21,23,27,33,51] 150 [3,11,19,31,33,51] 152 [3,7,9,17,37,51] 162 [3,17,21,23,37,51] 155 [3,7,23,31,37,51] 153 [17,19,27,31,37,51] 157 [9,17,19,33,37,51] 163 [7,9,17,23,39,51] 150 [3,11,23,33,39,51] 165 [9,11,19,37,39,51] 155 [9,13,23,37,39,51] 152 [1,3,7,27,41,51] 153 [1,19,29,31,41,51] 154 [3,9,11,33,41,51] 156 [9,11,23,37,41,51] 153 [1,9,27,37,41,51] 161 [13,21,27,37,41,51] 151 [9,21,27,39,41,51] 155 [1,7,19,27,43,51] 163 [1,9,27,29,43,51] 153 [7,9,11,37,43,51] 155 [1,11,17,19,47,51] 152 [3,13,21,29,47,51] 156 [13,27,31,33,47,51] 151 [19,23,31,37,47,51] 152 [19,21,23,39,47,51] 166 [7,9,13,23,49,51] 151 [7,11,19,29,49,51] 158 [1,13,21,29,49,51] 153 [7,19,27,31,49,51] 150 [3,21,33,37,49,51] 150 [9,27,33,37,49,51] 150 [17,31,37,41,49,51] 152 [1,17,23,47,49,51] 162 [1,7,9,13,21,53] 167 [1,7,11,19,27,53] 152 [3,9,11,21,27,53] 152 [3,11,19,23,31,53] 153 [1,9,13,27,33,53] 151 [3,7,17,29,33,53] 150 [3,19,27,29,33,53] 153 [1,9,11,23,37,53] 156 [1,7,11,23,39,53] 150 [1,13,23,29,39,53] 155 [1,3,27,29,39,53] 159 [1,11,31,37,39,53] 151 [1,17,31,37,39,53] 159 [9,23,33,37,39,53] 151 [7,9,23,27,41,53] 150 [11,17,23,39,41,53] 153 [13,21,27,39,41,53] 159 [3,11,31,39,41,53] 152 [3,21,23,27,43,53] 151 [1,17,21,29,43,53] 151 [9,11,29,33,43,53] 156 [3,21,31,33,43,53] 154 [9,11,17,37,43,53] 155 [1,19,29,39,43,53] 153 [11,23,37,39,43,53] 153 [11,17,19,41,43,53] 153 [1,3,7,19,47,53] 150 [7,13,17,21,47,53] 151 [1,3,27,29,47,53] 156 [1,19,31,37,47,53] 165 [9,11,19,39,47,53] 150 [1,19,37,39,47,53] 155 [1,9,31,43,47,53] 155 [23,29,37,43,47,53] 155 [3,17,21,27,49,53] 155 [7,21,27,39,49,53] 152 [7,19,27,47,49,53] 155 [19,21,37,47,49,53] 154 [3,7,41,47,49,53] 150 [1,7,17,19,51,53] 156 [3,7,11,27,51,53] 151 [3,7,19,31,51,53] 152 [7,11,23,37,51,53] 159 [11,19,23,37,51,53] 156 [7,17,37,41,51,53] 152 [9,23,37,41,51,53] 159 [1,23,37,43,51,53] 151 [1,11,19,47,51,53]
2012-04-03, 23:06   #7
bcp19

Oct 2011

10101001112 Posts

Quote:
 Originally Posted by axn These are the 164 primes for 3,7,17,21,43,49. Not sure your problem specification allows for 49 to come in the last place.
As sm_88 said, as long as 1 number is below 46 it works. 3,7,17,21,49,43 is the same as what you have, but in a different order. I notice you have no even numbers in any of your solutions, did you only do odd numbers or something? I did see a 167 in there, which beats the 159 I just ran across.

Also, you seem to have missed 1,3,9,17,53,3 which was my 159. I don't think you realized that the 6th number can be a duplicate.

Last fiddled with by bcp19 on 2012-04-03 at 23:11

2012-04-04, 01:01   #8
axn

Jun 2003

23×673 Posts

Quote:
 Originally Posted by bcp19 As sm_88 said, as long as 1 number is below 46 it works. 3,7,17,21,49,43 is the same as what you have, but in a different order. I did see a 167 in there, which beats the 159 I just ran across.
There is also a 169 there

Quote:
 Originally Posted by bcp19 Also, you seem to have missed 1,3,9,17,53,3 which was my 159. I don't think you realized that the 6th number can be a duplicate.
Hmmm.. Do you by chance count the same prime twice? BTW, quick check gives 126/2=63 primes for 1,3,9,17,53,3

Quote:
 Originally Posted by bcp19 I notice you have no even numbers in any of your solutions, did you only do odd numbers or something?
I wasn't doing exhaustive search -- merely "good" enough. The strategy was that I didn't want any combinations that would narrow down the potential combinations too much. That means no duplicates, and no numbers that end in 0,2,4,6,8,5. This cut down the search space so that I could do an exhaustive search.

2012-04-04, 03:12   #9
bcp19

Oct 2011

7×97 Posts

Quote:
 Originally Posted by axn There is also a 169 there Hmmm.. Do you by chance count the same prime twice? BTW, quick check gives 126/2=63 primes for 1,3,9,17,53,3 I wasn't doing exhaustive search -- merely "good" enough. The strategy was that I didn't want any combinations that would narrow down the potential combinations too much. That means no duplicates, and no numbers that end in 0,2,4,6,8,5. This cut down the search space so that I could do an exhaustive search.
Hmmm, I need to check my transposition worksheet then, I had set up 720 rows where I insert the 6 numbers in the top row and it automatically transposed the digits for all combinations, the 7th column cell formula multiplied the first 6 to get the number, the the IsPrime function I defined would produce 1 for true and 0 for false, so a simple sum gave me total. I did not stop to think about the double # creating 1/2 the primes, but that definitely shows something is wrong if I have a odd number of primes from a double number.

2012-04-06, 11:07   #10
ATH
Einyen

Dec 2003
Denmark

3,347 Posts

Quote:
 Originally Posted by bcp19 With the recent lotto fever, I got to thinking about a puzzle. Given 6 lotto numbers, 5 ranged 1-56 and 1 ranged 1-46(010203040501 to 525354555646)
I understand this as the 1-46 number can be a duplicate of one of the 5 1-56 numbers? If this is correct the highest one is 204 primes with: 3,19,23,31,33 and 33
2nd highest is 194 primes with: 7,9,31,37,49 and 7.

It took 19 hours to search the entire 46*56*55*54*53*52/(1*2*3*4*5) = 175,711,536 combinations with a quick program not really optimized and while gaming on the computer as well.

If duplicates is not allowed the 169 primes from axn is the highest.

2012-04-06, 14:03   #11
axn

Jun 2003

23·673 Posts

Quote:
 Originally Posted by ATH I understand this as the 1-46 number can be a duplicate of one of the 5 1-56 numbers? If this is correct the highest one is 204 primes with: 3,19,23,31,33 and 33 2nd highest is 194 primes with: 7,9,31,37,49 and 7.
The last one can be a duplicate. But the question is, are duplicate primes allowed? Because, out of the 204 (or 194) primes, only 102 (resp 97) are unique.

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