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I see. Excellent - you have pretty much solved it, except my answer is 7 fewer than yours, because I meant that there is only one white, one yellow,..., one black - not two of each. (I'm not sure why you thought two - perhaps my wording "one each of a white ball, yellow, green, brown, blue, pink and black balls" is a bit ambiguous, ending as it does in "balls"? But how do you expect to play with two white balls - this is Nookers, not Billiards!)
Anyway, subtract 7 from your answer and we agree (except, you have not quite proved that your answer is the only one possible, which it is)!
So, on to question 5:
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Q5)
Around the time of the 200,007 Crucibilis Nookers Plus Final, one of Crucibilis's best players, Sonny O'Jollyman had got bored with potting his shrubs and was playing a tournament of 962,537,588,106-Ball pool. This delightful game is played on a smiliar table to a Nookers table, only smaller, but with bigger pockets - pockets the size of Pluto (the planet-and-soon-possibly-not-to-be-a-planet, that is, not the cartoon dog) - and the same number of balls (some the size of grapefruit, some the size of ripe tomatoes).
Each ball, except the white, is numbered from 1 to 962,537,588,106, which the players have to strike (with the white) in order, the winner being the one who pots the 962,537,588,106-ball.
While Sonny was about to play his long-time best pal, Ebbie Petdon, in the Final, it was the job of the delightful referee, Tabitha Michaels, to keep the balls clean, something which the spectators loved to watch. Unfortunately, with so many balls to clean, her gloves got rather dirty and it so happened that, while wiping a ball that was dirty made it clean, when she wiped a ball that was already clean she actually made it dirty!
At the start of the Final, all the balls were dirty (as Haggis Allixins had dropped beer on them during his thrilling comeback against Jammy Brown in the Semi-Final). So Tabitha first wiped every ball and made it clean. (She never wiped the white ball again.)
Ebbie wasn't satisfied, so asked that she clean some again. In fact, she then wiped only every ball divisible by 2. Sonny didn't think this was sufficient, so asked that she wipe some again, and this time she then wiped every ball divisible by 3. In fact, this continued, with her then wiping every ball divisible by 4, then every ball divisible by 5,..., through to 962,537,588,106.
By the time she had finished, the audience had gone home, because much as they enjoyed watching her polishing the balls, enough was enough, and the match was never played.
However, if the players had then started, how many clean balls would there have been on the big 962,537,588,106-Ball table?
Anyway, subtract 7 from your answer and we agree (except, you have not quite proved that your answer is the only one possible, which it is)!
So, on to question 5:
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Q5)
Around the time of the 200,007 Crucibilis Nookers Plus Final, one of Crucibilis's best players, Sonny O'Jollyman had got bored with potting his shrubs and was playing a tournament of 962,537,588,106-Ball pool. This delightful game is played on a smiliar table to a Nookers table, only smaller, but with bigger pockets - pockets the size of Pluto (the planet-and-soon-possibly-not-to-be-a-planet, that is, not the cartoon dog) - and the same number of balls (some the size of grapefruit, some the size of ripe tomatoes).
Each ball, except the white, is numbered from 1 to 962,537,588,106, which the players have to strike (with the white) in order, the winner being the one who pots the 962,537,588,106-ball.
While Sonny was about to play his long-time best pal, Ebbie Petdon, in the Final, it was the job of the delightful referee, Tabitha Michaels, to keep the balls clean, something which the spectators loved to watch. Unfortunately, with so many balls to clean, her gloves got rather dirty and it so happened that, while wiping a ball that was dirty made it clean, when she wiped a ball that was already clean she actually made it dirty!
At the start of the Final, all the balls were dirty (as Haggis Allixins had dropped beer on them during his thrilling comeback against Jammy Brown in the Semi-Final). So Tabitha first wiped every ball and made it clean. (She never wiped the white ball again.)
Ebbie wasn't satisfied, so asked that she clean some again. In fact, she then wiped only every ball divisible by 2. Sonny didn't think this was sufficient, so asked that she wipe some again, and this time she then wiped every ball divisible by 3. In fact, this continued, with her then wiping every ball divisible by 4, then every ball divisible by 5,..., through to 962,537,588,106.
By the time she had finished, the audience had gone home, because much as they enjoyed watching her polishing the balls, enough was enough, and the match was never played.
However, if the players had then started, how many clean balls would there have been on the big 962,537,588,106-Ball table?
). It's clear a can't be odd, because then you'd have odd over even, but other than that, I'm stumped. I'll have another look tomorrow.
(With their gnarled hands you wouldn't be able to see where they were pointing, anyway.)
Would you care to give a proof?
,I never learned math in English and I have serious doubts about the words I have to use.
Anyway,if the ball was clean or dirty in the end depends on how many times it was wiped. Those balls ,which were wiped odd number of times (1, 3, 5...) ,should be clean and those ,which were wiped even number of times ,should be dirty. The number of wipings depends on how many factors does a number have (I mean, for example number 10 has 4 factors - 1, 2, 5 & 10, ball number 10 will be wiped 4 times and will be dirty, number 25 has 3 factors - 1, 5 & 25 , ball number 25 will be wiped 3 times and will be clean, etc.) Most numbers have even number of factors ( they are forming a pairs(at least they do it in my country)), exceptional are numbers which are squares of some other number (1, 4, 9, 16...100, 121, 144...etc.)The biggest square number less than 962537588106 is 962537588100, which is square of 981090 ( I just took the square root of the number of balls).It means that there are 981090 square numbers, it is 981090 clean balls plus white ball,which is clean too.
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