HP Forum Archive 19

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an interesting challenge Message #1 Posted by Don Shepherd on 10 Aug 2010, 9:18 p.m. Now here is something very interesting. I'm not sure it requires a program to calculate, although you may try. What is the highest possible score in a standard game of bowling if your score in each of the ten frames is required to be a prime number?

Re: an interesting challenge Message #2 Posted by Karl Schneider on 10 Aug 2010, 11:27 p.m., in response to message #1 by Don Shepherd Quote: What is the highest possible score in a standard game of bowling if your score in each of the ten frames is required to be a prime number? Note to international readers: this is 10-pin American bowling, not German 'Kegeln' or other game... The player throws twice in each of ten frames, or until all ten pins are knocked down, whichever comes first. For a 'strike' (all pins down in one throw), the player gets to add the number of pins knocked down in the next two throws. For a 'spare' (all pins down in two throws), the player gets to add the number of pins knocked down in the next throw. So, a score in one frame may be contingent on the success in subsequent frames. A perfect game is 12 strikes, for a score of 300, because up to two extra throws can be allowed in frame 10. There's the trivial solution of a seven in every frame for a total of 70, but we can do better... Assume strikes in frames 1 and 2, followed by a nine and a miss (zero) in frame three. This gives 29 and 19 as the scores in frames 1 and 2 (excellent!), but leaves a non-prime nine in frame 3. A nine as the first throw in frame 3 leaves no acceptable outcome, so we must make the next-largest prime: 23. Now assume strikes in frames 1 and 2, followed by a three and a four in frame 3. This gives 23 and 17 as the scores in frames 1 and 2, with a seven in frame 3 -- all primes. Repeat that sequence twice for frames 4-9, then two strikes and a nine in frame 10: 3(23 + 17 + 7) + 29 = 170. That's my best answer... -- KS Edited: 11 Aug 2010, 11:48 p.m. after one or more responses were posted

Re: an interesting challenge Message #3 Posted by Don Shepherd on 10 Aug 2010, 11:33 p.m., in response to message #2 by Karl Schneider Thanks, Karl. According to my source, that is not the correct answer. I'll leave it at that for now. Don

Re: an interesting challenge Message #4 Posted by Don Shepherd on 10 Aug 2010, 11:38 p.m., in response to message #2 by Karl Schneider And the 10th frame final score of 170 is not a prime number!

Re: an interesting challenge Message #5 Posted by Bart (UK) on 11 Aug 2010, 10:31 a.m., in response to message #4 by Don Shepherd You said: "score in each of the ten frames" means the score of each individual frame (which still would depend on the consecutive frame depending on spare/strike). But if you had said: "the ten frame scores" or "the scores of the ten frames" it could be interpreted as the score at the end of each frame, i.e. cumulative, as is the norm of the scoring. "each of" implies individual, don't complain when people interpret correctly what you say rather than what you think you are saying. Although the second way of stating it could still be open to mis-interpretation, you could correct those less familiar with scoring norms without exclamation marks and examples of how wrong they are.

Re: an interesting challenge Message #6 Posted by Don Shepherd on 11 Aug 2010, 12:22 p.m., in response to message #5 by Bart (UK) Bart, I used the same language that was used in the source of this riddle. I assumed that those who know about how bowling scores are calculated know that "score" = cumulative score, not the pins knocked down in an individual frame. Bowling score sheets have a little box at the top of each frame where you record either the pins knocked down (on each of two throws) or strike (X) or spare(/). But then, when you can--and this might be two frames later--you write the final cumulative score in the bigger part of the box for each frame, at the bottom. So if you realize that "score" = cumulative score, then the language "score in each of the ten frames" means the cumulative score as recorded in each frame, and those 10 numbers must all be prime. My post about 170 not being a prime number was written before I realized that Karl had interpreted "score" as an individual frame score, which was not the intent. And I didn't "complain." Don't interpret an exclamation point as a "complaint." I merely pointed out that 170 was not prime and it has to be given what "score" means in this context. Don

Re: an interesting challenge Message #7 Posted by Bart (UK) on 12 Aug 2010, 5:16 a.m., in response to message #6 by Don Shepherd Quote: I assumed that those who know about how bowling scores are calculated know that "score" Your replies came after Karl's message clearly highlighted that those outside the US may not be that familiar with the game. Quote: My post about 170 not being a prime number was written before I realized that Karl had interpreted "score" as an individual frame score, which was not the intent. It was followed by a terse one liner that "cumulative" was the norm and a reference to a single obcure picture of a scoring card, rather than - as Karl did - a link to the whole article so as to benefit the wider international audience of the forum. It may or may not have been your intent, but your messages cumulatively did come across as "stupid man, do you not know that". May I kindly remind you that there is a "rest of the world" outside the US.

Re: an interesting challenge Message #8 Posted by Don Shepherd on 12 Aug 2010, 7:42 a.m., in response to message #7 by Bart (UK) Bart, if you were offended by anything that I wrote, and apparently you were, then I'm sorry. But, honestly, I don't care. My purpose in posing this problem was not to educate the rest of the world on how bowling is done. It was to offer an interesting problem related to mathematics. And two people were able to arrive at the final solution, and I congratulate them, and I hope that some of us learned something in this little exercise. Don

Re: an interesting challenge Message #9 Posted by Mike on 11 Aug 2010, 5:04 p.m., in response to message #5 by Bart (UK) Someone said: "score in each of the ten frames" You replied: means the score of each individual frame (which still would depend on the consecutive frame depending on spare/strike). I reply: You don't bowl, do you. The score "in each frame" has been the accumulated total, since the game was invented. :-)

Re: an interesting challenge Message #10 Posted by Bart (UK) on 12 Aug 2010, 5:38 a.m., in response to message #9 by Mike Now your message really comes across as "stupid man, do you not know that". May I kindly remind you also that there is a "rest of the world" outside the US. By the way, I have bowled (not as a regular, but 2-3 times a year), and though I may not be good at it I do know how it works (here in the UK it's the same). However, it was clear from Karl's first post that perhaps many don't, but would still like to attempt the challenge. A simple explanation and a link to the article as in Karl's second post would have been so much more beneficial rather than a few terse sentences and a link just to an obscure picture of a scorecard. Now, I particularly like the article on scoring primes, certainly a challenge for my next game.

Re: an interesting challenge Message #11 Posted by Martin Pinckney on 12 Aug 2010, 9:29 a.m., in response to message #10 by Bart (UK) Bart, with all due respect, in this interchange between you and Don, I for one did not read anything condescending in Don's original messages. Explaining bowling scoring to "the rest of the world" is another matter. Perhaps it's a language thing. After all, we are "two people separated by a common language." Don, saying "I'm sorry... but I really don't care" is contradictory, shall we say? My contribution to international relations for today. :-)

Re: an interesting challenge Message #12 Posted by Don Shepherd on 12 Aug 2010, 10:23 a.m., in response to message #11 by Martin Pinckney Thanks Martin. Contradictory? I considered that possibility, but here are my thoughts. I'm sorry that anyone would or could interpret things I wrote as offensive to anyone because, as you pointed out, that certainly was not my intent. I thought the language I used would be perfectly clear to anyone familiar with bowling. Everyone should know that we're talking about the cumulative score as listed in each frame. And so in my original post I used the exact language from the book where I first read about this riddle. I would not have predicted that this brouhaha would have ensued, and I'm sorry that it did too. And I would think that a "terse" explanation would be better than a lengthy explanation, but if someone doesn't see it that way, and reads offense in it, well I'm sorry about that too. But, ultimately, I'm not going to care if someone misinterprets or feels otherwise offended because there are much bigger problems in the world to be concerned about, and life's too short to dwell on things like this. Hope this helps. I'm beginning to understand why we don't hear from our friend V anymore. Don

Re: an interesting challenge Message #13 Posted by Bart (UK) on 13 Aug 2010, 5:27 a.m., in response to message #12 by Don Shepherd Quote: there are much bigger problems in the world I.e. in your small world that doesn't care about the 90% that don't know all the ins and outs of ten pin bowling. Quote: I thought the language I used would be perfectly clear to anyone familiar with bowling. Everyone should know that we're talking about the cumulative score as listed in each frame. Sorry, I apologize, I mis-understood. Your challenge was only for those that know bowling.

Re: an interesting challenge Message #14 Posted by Gerson W. Barbosa on 12 Aug 2010, 10:47 a.m., in response to message #11 by Martin Pinckney Quote: My contribution to international relations for today. :-) That's a delicate matter, but it appears the British are concerned about that :-) http://edition.cnn.com/2010/WORLD/europe/08/12/uk.olympic.etiquette/#fbid=c-Pf3d_Q1Ao&wom=false

Re: an interesting challenge Message #15 Posted by Don Shepherd on 10 Aug 2010, 11:41 p.m., in response to message #2 by Karl Schneider The score in each frame is a cumulative number, so frames 1, 2, and 3 might be, for example, 13, 23, 37, and so on. Don

Re: an interesting challenge Message #16 Posted by Karl Schneider on 11 Aug 2010, 12:02 a.m., in response to message #15 by Don Shepherd Quote: The score in each frame is a cumulative number, so frames 1, 2, and 3 might be, for example, 13, 23, 37, and so on. Then you should have stated, "cumulative score". It's been a long time since I bowled, but don't the traditional score sheets have a place in each frame for the frame score and the running total? This facilitates manual addition, particularly in the era before automatic scoring. (The pinsetting machine was invented only in 1962.) Accumulating the score as you go makes sense for convenience, and to see who's ahead. A prime-valued cumulative score makes for a much more difficult challenge -- probably best suited for a computer program. -- KS

Re: an interesting challenge Message #17 Posted by Don Shepherd on 11 Aug 2010, 12:13 a.m., in response to message #16 by Karl Schneider Here is an example of a bowling score sheet. The cumulative score goes in the big area at the bottom, those are the numbers that have to be prime. Don

Re: an interesting challenge Message #18 Posted by Karl Schneider on 11 Aug 2010, 1:06 a.m., in response to message #17 by Don Shepherd Don -- The webpage where you apparently obtained that particular photo also lists frame scores under "Breaking Down a Sample Score". Alas, of the scoring sheets offered at the sites found by web-search, I did not see any that provided a designated place to write frame scores. This may explain why arithmetic-challenged folks scribbled in the margins when necessary. Having frame scores (intermediate results) written down would have helped prevent errors in the old days of manual scoring -- a lesson on the virtues of RPN! Another lesson pertaining to calculators and other electronic ADP equipment: Some of these sites discuss how people are forgetting (or never learned) how to score manually, because the automated systems do everything for you... -- KS Edited: 11 Aug 2010, 1:10 a.m.

Re: an interesting challenge Message #19 Posted by Don Shepherd on 11 Aug 2010, 7:51 a.m., in response to message #18 by Karl Schneider You are right, the "frame scores" were typically not written on the standard bowling score sheets. I think the cumulative score was deemed much more important, so that the current leader would always be known. I'm sure many newbies to bowling today have no idea how the score is calculated, since all the bowling alleys I have been in lately use automated scoring equipment. When I was growing up in the 50's and 60's, bowling was big; my mother was always in a league at the local alley, and everyone learned how to keep score. And if you were serious about it, you bought your own bowling ball and shoes (I've still got mine from around 1970; alas, the shoes don't fit anymore). I took my 8th grade math class bowling a couple of years ago (very small school, only 3 girls in the 8th grade). They had bowled before, and loved it. Don

Re: an interesting challenge Message #20 Posted by Karl Schneider on 11 Aug 2010, 11:31 p.m., in response to message #19 by Don Shepherd Quote: You are right, the "frame scores" were typically not written on the standard bowling score sheets. I think the cumulative score was deemed much more important, so that the current leader would always be known. I noticed that some of the template score sheets provided two small boxes for recording the throws and a generous amount of space in each frame; others were compact and offered only one small box. I'll bet that many folks wrote down the frame score in small numerals on the left side of the frame box, space permitting. That might be helpful for mid-game scores like, e.g., 86 + 27 = 113. Yes, cumulative summing keeps the players abreast of the situation, but the scores of good bowlers are always a frame or two behind in tabulation. The more important reason for as-you-go cumulative summation is so that the scorer didn't have to add up all the frames at the end, without benefit of an adding machine or calculator. This brings to mind several features -- which may likely be a reality -- of automatic scoring: 1. An option to hide the frame and cumulative scores, then compute final scores at the end of the game, leaving the identity of the winner a possible surprise. 2. Showing the minimum and maximum cumulative scores for a frame (pending subsequent throws) in a different color, until the results are complete. -- KS Edited: 12 Aug 2010, 1:08 a.m.

Re: an interesting challenge Message #21 Posted by Don Shepherd on 12 Aug 2010, 7:29 a.m., in response to message #20 by Karl Schneider Quote: The more important reason for as-you-go cumulative summation is so that the scorer didn't have to add up all the frames at the end, without benefit of an adding machine or calculator. Karl, as another poster pointed out, bowling scores have always been done this way, with the cumulative score being the prominent feature of each frame. In league play, before the advent of automated scoring, most alleys had projectors that displayed the handwritten entries on a screen above the alley, for all to see. Bowling was very competitive, and everyone wanted to see which team was ahead of the other. Sometimes bets were made, and there was keen interest in the current score. The little box (or, as you say, sometimes boxes) at the top were really there only for the purpose of calculating the cumulative score, especially for strikes and spares. And every bowler knew how to keep score. For many, probably, it was the only math they used on a regular basis. In recent years, with the automated scoring system, I've only been bowling about 3 times, so I can't speak to the options available in those automatic scoring systetms. But I'd be very surprized if they have an option to suppress the cumulative score until the end of the game. Imagine a football or basketball game where the scoreboard was blank until the end of the game. I think the fans would revolt! Thanks for participating in this little exercise. Don

Re: an interesting challenge Message #22 Posted by Karl Schneider on 14 Aug 2010, 3:36 a.m., in response to message #21 by Don Shepherd Quote: Karl, as another poster pointed out, bowling scores have always been done this way, with the cumulative score being the prominent feature of each frame. In league play, before the advent of automated scoring, most alleys had projectors that displayed the handwritten entries on a screen above the alley, for all to see. I am not unfamiliar with any of this. I've a bowled a few times (decades ago), and I've been inside bowling alleys over the years. Next opportunity, I'll take a closer look at the scoring technology inside the alley I pass by on a routine basis. Quote: Bowling was very competitive, and everyone wanted to see which team was ahead of the other. Sometimes bets were made, and there was keen interest in the current score. Of course, in the case of good bowlers, that 'current score' is rarely known precisely, due to the need for adding the results of one or two pending throws. Still, it would make little sense not to accumulate scores 'as you go', especially when the arithmetic was done manually. Quote: The little box (or, as you say, sometimes boxes) at the top were really there only for the purpose of calculating the cumulative score, especially for strikes and spares. Hmm? They are there for recording the results of the individual throws... Quote: ...I can't speak to the options available in those automatic scoring systems. But I'd be very surprised if they have an option to suppress the cumulative score until the end of the game. Imagine a football or basketball game where the scoreboard was blank until the end of the game. I think the fans would revolt! I'd also thought of the (American) football or basketball comparison: Football and basketball are clock-driven, and a team's strategy and tactics as the clock winds down will be influenced by the ahead-or-behind standing. Bowling, like baseball, is not clock-driven. Also, I doubt that a player's objective would normally depend on the score: just try to roll strikes and spares each time. (Knowing he's behind, though, may put some pressure on a player.) -- KS Edited: 14 Aug 2010, 3:37 a.m.

Re: an interesting challenge Message #23 Posted by Don Shepherd on 14 Aug 2010, 7:01 p.m., in response to message #22 by Karl Schneider Quote: I doubt that a player's objective would normally depend on the score: just try to roll strikes and spares each time. I fully agree. A bowler's strategy is to get as many strikes as possible. Failing that, spares. Failing that, just knock down as many pins as you can. That is the strategy, regardless of the current score. But the goal is to beat the other players (if you are in a league, especially, and especially if there is money riding on it, which is not uncommon). And to beat the other players, you need to know where you, and they, currently stand. Hence, cumulative score. If you're bowling for fun, which is what I have always done, your goal is just to have a good time. Don

Re: an interesting challenge Message #24 Posted by Chuck on 11 Aug 2010, 12:53 a.m., in response to message #1 by Don Shepherd Well, I get 181 with the following sequence: 23 43 53 61 89 109 127 137 157 181 Any higher than that?

Re: an interesting challenge Message #25 Posted by Don Shepherd on 11 Aug 2010, 7:42 a.m., in response to message #24 by Chuck Chuck, my source shows a higher score than that. Don

Re: an interesting challenge Message #26 Posted by Alex L on 11 Aug 2010, 10:54 a.m., in response to message #25 by Don Shepherd Here's the result of manually applying a greedy algorithm... X X 3/ 8/ 8/ 8/ 6/ 8/ X XXX 23 43 61 79 97 113 131 151 181 211 Curious to see if there are improvements.

Re: an interesting challenge Message #27 Posted by Don Shepherd on 11 Aug 2010, 12:05 p.m., in response to message #26 by Alex L Alex, a higher score is possible. Don

Re: an interesting challenge Message #28 Posted by Alex L on 11 Aug 2010, 1:02 p.m., in response to message #27 by Don Shepherd I figured more might be possible, looking heuristically for runs of primes separated by 30 & 20... -/ 7/ X X X X X 42 -/ XXX 17 37 67 97 127 151 167 173 193 223 The two gutter balls can actually be any rolls.

Re: an interesting challenge Message #29 Posted by Bill (Smithville, NJ) on 11 Aug 2010, 1:27 p.m., in response to message #28 by Alex L Quote: The two gutter balls can actually be any rolls. But I really like the idea that you can start with a gutter ball and still end up with a high score. Bill

Re: an interesting challenge Message #30 Posted by Don Shepherd on 11 Aug 2010, 1:35 p.m., in response to message #28 by Alex L You're getting there..... : ) Don

Re: an interesting challenge Message #31 Posted by Alex L on 11 Aug 2010, 1:58 p.m., in response to message #30 by Don Shepherd I'm actually enjoying working this problem manually. But I have to think that with 6 consecutive strikes and no open frames, I've got to be close. At least I've narrowed down any brute force search. :) -/ 7/ X X X X X X 2/ 2/8 17 37 67 97 127 157 179 199 211 229 ** after posting, I see that Jeff O. beat me to this score. ** Edited: 11 Aug 2010, 2:03 p.m.

Re: an interesting challenge Message #32 Posted by Bill (Smithville, NJ) on 11 Aug 2010, 8:59 a.m., in response to message #1 by Don Shepherd Don, Very interesting challenge. What's interesting is the question "do you want to maximize the first frame score", or would a lower score in the first frame ultimatelly result in a higher total score. From a calculation point, I'm thinking a brute force recursive method would solve it on a computer, but wouldn't work on a calculator. For the mathematical bowlers, you might want to try the following bowling game next time at the lanes: Rules for Bowling for Primes Bill

Re: an interesting challenge Message #33 Posted by Katie Wasserman on 11 Aug 2010, 10:22 a.m., in response to message #32 by Bill (Smithville, NJ) I like it, but the scoring is a little complex and hard to remember. Maybe we can convince Nintendo to add this option to their Wii bowling games. I'm sure the retired mathematicians in the Wii bowling league will appreciate this. For brick and mortar bowlers, this would be a perfect program to put on your favorite, programmable HP calculator.

Re: an interesting challenge Message #34 Posted by Ken Shaw on 11 Aug 2010, 12:59 p.m., in response to message #1 by Don Shepherd This is starting to look hard. If this problem is extended to an arbitrary number of frames (greater than 10), could someone here figure out if it's another one of those NP-complete problems? Maybe we can get the Clay Institute to offer up $1,000,000 :-)

Re: an interesting challenge Message #35 Posted by Alex L on 11 Aug 2010, 1:38 p.m., in response to message #34 by Ken Shaw Once the score reaches 1327, there's a prime gap of 34, so no next frame score is possible (Wikipedia - Prime Gap). So there's an upper bound on the number of frames.

Re: an interesting challenge Message #36 Posted by Ken Shaw on 11 Aug 2010, 2:27 p.m., in response to message #35 by Alex L I guess that was obvious. Oops.

Re: an interesting challenge Message #37 Posted by Jeff O. on 11 Aug 2010, 1:41 p.m., in response to message #1 by Don Shepherd How about: Frame: 1 2 3 4 5 6 7 8 9 10 Pins: 7- X X X X X X X 2/ 2/8 Score: 7 37 67 97 127 157 179 199 211 229

Re: an interesting challenge Message #38 Posted by Don Shepherd on 11 Aug 2010, 2:17 p.m., in response to message #37 by Jeff O. Jeff and Alex, congratulations, you got it. You are steely-eyed misslemen, or at least math whizzes. My source.

Re: an interesting challenge Message #39 Posted by Jeff O. on 11 Aug 2010, 3:38 p.m., in response to message #38 by Don Shepherd Well, I don't think that I "devised an ingenious solution to a tough problem under extreme pressure." There was no pressure, other than the glory of being first to get the correct answer in a very clever group. Not a math whiz either, got it more or less by trial and error. Figuring the highest score would start with two strikes, I then tried "rolls" that continued to give prime numbers. If I hit a dead end, I back-tracked and tried again. The best I could do starting with two strikes was 193, which did not seem very high. Then I tried starting with one strike and came up with 193 again. I thought I tried starting with a spare, but if so I must have made an error because as Alex demonstrated, doing so can lead to the correct answer. I ended up going with an open first frame with the highest score possible, 7, and went from there. Helped a lot that 37, 67, 97, 127, and 157 are all primes, as seven strikes in a row does a lot to get a good score. Regarding your source, great, another book about prime numbers that I will have to buy. I already have "Music of the Primes" and "Prime Obsession" (which is an awesome book that didn’t make me feel too stupid.) I'm sure there many, many more books about prime numbers, but I only have so much time to read such things. Edited: 11 Aug 2010, 3:40 p.m.

Re: an interesting challenge Message #40 Posted by Don Shepherd on 11 Aug 2010, 4:19 p.m., in response to message #39 by Jeff O. Thanks, Jeff. As you and Alex showed, there is more than one solution (first frame 7, first frame 17). The book's solution is first frame 17. The book's authors claim that 229 is the highest score possible, and I was wondering if anyone here in this august group might find a higher score, and we could write the authors and tell them they are wrong. I thought it was an interesting problem, if not earth-shaking. It does make you do one thing that seems to have diminished importance today, and the word was an IBM motto for many years: think. Congratulations to you and Alex again (and everyone who contributed) and thanks for participating. Edited: 11 Aug 2010, 4:20 p.m.

Re: an interesting challenge Message #41 Posted by Egan Ford on 11 Aug 2010, 5:24 p.m., in response to message #40 by Don Shepherd I would argue that Alex's 223 solution is the practical highest since the 229 solutions (there is more than one) requires that 2 pins be knocked down on the first ball. Near impossible. Maybe that can be the HHC 2010 challenge--knock down 2 pins on the first ball. Edited: 11 Aug 2010, 5:24 p.m.

Re: an interesting challenge Message #42 Posted by Don Shepherd on 11 Aug 2010, 5:34 p.m., in response to message #41 by Egan Ford Near impossible? Bowl with me sometime and you'll change your mind! : ) Don

Re: an interesting challenge Message #43 Posted by Palmer O. Hanson, Jr. on 11 Aug 2010, 9:38 p.m., in response to message #42 by Don Shepherd Don: In response to the statement that getting two pins on the first ball was nearly impossible you wrote Quote: Near impossible? Bowl with me sometime and you'll change your mind! I think getting two on the first ball is about as likely as a golfer getting a hole-in-one. It is particularly unlikely for a bowler who has just got the string of strikes in the solution. While I can't recall seeing a two on a first ball I can recall seeing a bowler pick up a seven-ten split but I can't recall if a rebound from the pit was involved. My bowling experience predates yours. My high school had an intramural coeducational bowling league in my senior year (1946). I was also on a team that competed (ineptly, in my case) interscholastically. I also did some pin setting. That was one of the best paying jobs for teen age boys in those days but it was a little dangerous. Palmer

Re: an interesting challenge Message #44 Posted by Don Shepherd on 11 Aug 2010, 10:12 p.m., in response to message #43 by Palmer O. Hanson, Jr. Quote: I think getting two on the first ball is about as likely as a golfer getting a hole-in-one. It is particularly unlikely for a bowler who has just got the string of strikes in the solution. You may be right, but I've had some pretty bad rolls in my day and I seem to recall my ball just barely hitting the 7 pin and the 7 pin then knocks over the 8 pin (or, on the other side, the 10 pin and 9 pin). I've also got many a 7-10 split for hitting the 1 pin headon, rather than aiming for the "pocket" which I've never been able to do consistently. I know I've never converted a 7-10 split into a spare, but many a time I've hit either the 7 or the 10! Yeah, the likelihood of getting a string of strikes and THEN getting a 2 would appear to be pretty small, I admit. And I've rolled many a gutter ball, but I'm too proud to ask for bumpers today!

Re: an interesting challenge Message #45 Posted by DavidShenk on 11 Aug 2010, 7:11 p.m., in response to message #40 by Don Shepherd Discussion of this problem can be found here.

Re: an interesting challenge Message #46 Posted by Martin Pinckney on 11 Aug 2010, 5:50 p.m., in response to message #39 by Jeff O. Hopeless geeks.

Re: an interesting challenge Message #47 Posted by Don Shepherd on 11 Aug 2010, 10:15 p.m., in response to message #46 by Martin Pinckney And proud of it! I still walk around with 80-column punchcards in my pocket sometimes. : )

Re: an interesting challenge Message #48 Posted by Jeff O. on 11 Aug 2010, 10:42 p.m., in response to message #46 by Martin Pinckney Guilty as charged.

Re: an interesting challenge Message #49 Posted by Gerson W. Barbosa on 12 Aug 2010, 7:57 a.m., in response to message #39 by Jeff O. Quote: Regarding your source, great, another book about prime numbers that I will have to buy. While you don't, you can take a look at the online version of his source. Since Don's previous challenge is a dead topic now I think disclosing his source, which I had already found, should be no problem: http://primes.utm.edu/curios/page.php?short=85837

Re: an interesting challenge Message #50 Posted by Don Shepherd on 12 Aug 2010, 8:31 a.m., in response to message #49 by Gerson W. Barbosa Yeah, thanks Gerson, I'd forgotten about that! Don

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