Oh yes. *winds neck in*Dave Ricesky wrote:Charlie's question was about using different sets of large numbers - and 100*99*98*97*10*10 is way too big, for example.Ian Volante wrote:Really? How, given that 25*50*75*100*10*10 is under 1x10^9 and 2^31 is >2x10^9? Some intermediate factoring steps?Dave Ricesky wrote:you can reach values larger than 2^31
How many Numbers puzzles are there in total?
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Re: How many Numbers puzzles are there in total?
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Re: How many Numbers puzzles are there in total?
I have an answer for you, Charlie. The easiest four numbers are:Charlie Reams wrote:Great topic, enjoyed reading all this!
Harder question: if one were designing a new variant, which four large numbers (in the range 11-100) make the game hardest (i.e. fewest games solvable) and which easiest (i.e. most games solvable)?
*** Drum roll ***
59, 83, 93, 97, with 11,072,459 of the 11,905,457 games solvable, or 9,108,733 / 9,343,307 if you discount 6 small. That's 93.00% or 97.49% respectively.
The hardest four numbers are 12, 16, 18, 24, with 10,510,350 / 11,905,457 (88.28%) or 8,546,624 / 9,343,307 (91.47%).
The breakdowns by number of larges chosen is:
59, 83, 93, 97
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 5,063,270 / 5,221,392 (96.97%)
2 large: 3,259,190 / 3,317,310 (98.25%)
3 large: 738,513 / 755,160 (97.80%)
4 large: 47,760 / 49,445 (96.59%)
12, 16, 18, 24
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 4,704,925/ 5,221,392 (90.11%)
2 large: 3,087,519 / 3,317,310 (93.07%)
3 large: 707,920 / 755,160 (93.74%)
4 large: 46,260 / 49,445 (93.56%)
25, 50, 75, 100 performs relatively poorly, coming 2,378,830th out of the 2,555,190 possible selections - that's in the 7th percentile.
It still beats 11, 12, 13, 14, which comes 2,554,989th out of 2,555,190, with 10,621,119 / 11,905,457 (89.21%) or 8,657,393 / 9,343,307 (92.66%)
97,98,99,100 is actually not too bad, coming 1,677,651th out of 2,555,190 (35th percentile), with counts of 10,933,717 / 11,905,457 (91.84%) or 8,969,991 / 9,343,307 (96.00%)
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Re: How many Numbers puzzles are there in total?
Where do 12, 37, 62, 87 fall?Dave Ricesky wrote:I have an answer for you, Charlie. The easiest four numbers are:Charlie Reams wrote:Great topic, enjoyed reading all this!
Harder question: if one were designing a new variant, which four large numbers (in the range 11-100) make the game hardest (i.e. fewest games solvable) and which easiest (i.e. most games solvable)?
*** Drum roll ***
59, 83, 93, 97, with 11,072,459 of the 11,905,457 games solvable, or 9,108,733 / 9,343,307 if you discount 6 small. That's 93.00% or 97.49% respectively.
The hardest four numbers are 12, 16, 18, 24, with 10,510,350 / 11,905,457 (88.28%) or 8,546,624 / 9,343,307 (91.47%).
The breakdowns by number of larges chosen is:
59, 83, 93, 97
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 5,063,270 / 5,221,392 (96.97%)
2 large: 3,259,190 / 3,317,310 (98.25%)
3 large: 738,513 / 755,160 (97.80%)
4 large: 47,760 / 49,445 (96.59%)
12, 16, 18, 24
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 4,704,925/ 5,221,392 (90.11%)
2 large: 3,087,519 / 3,317,310 (93.07%)
3 large: 707,920 / 755,160 (93.74%)
4 large: 46,260 / 49,445 (93.56%)
25, 50, 75, 100 performs relatively poorly, coming 2,378,830th out of the 2,555,190 possible selections - that's in the 7th percentile.
It still beats 11, 12, 13, 14, which comes 2,554,989th out of 2,555,190, with 10,621,119 / 11,905,457 (89.21%) or 8,657,393 / 9,343,307 (92.66%)
97,98,99,100 is actually not too bad, coming 1,677,651th out of 2,555,190 (35th percentile), with counts of 10,933,717 / 11,905,457 (91.84%) or 8,969,991 / 9,343,307 (96.00%)
I'm not dead yet. In a rut right now because of stress from work. I'll be back later in S89. I also plan to bring back the Mastergram - if I can find a way to run a timer or clock through pure MediaWiki without having to upload to Vimeo every time.
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Re: How many Numbers puzzles are there in total?
2 prime numbers, and 62 having 31 (a prime greater than 10) as a factor should put it much higher than the default.Johnny Canuck wrote:Where do 12, 37, 62, 87 fall?
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Re: How many Numbers puzzles are there in total?
Fantastic research, it goes without saying.
I think I'm right in saying that the selection which makes the game hardest (fewest games solvable) is 1,1,2,2,3,3. This is also the only selection where it is impossible not to max the round whatever the target.
I think I'm right in saying that the selection which makes the game hardest (fewest games solvable) is 1,1,2,2,3,3. This is also the only selection where it is impossible not to max the round whatever the target.
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Re: How many Numbers puzzles are there in total?
87 = 3*29, but anyway...Thomas Cappleman wrote:2 prime numbers, and 62 having 31 (a prime greater than 10) as a factor should put it much higher than the default.Johnny Canuck wrote:Where do 12, 37, 62, 87 fall?
12, 37, 62, 87 comes 1,810,325th out of 2,555,190 (30th percentile, so worse than 97, 98, 99, 100) with a breakdown as follows:
12, 37, 62, 87
----------------------------
6 small: 1,963,726 / 2,562,150 (76,64%)
1 large: 4,906,599 / 5,221,392 (93.97%)
2 large: 3,257,138 / 3,317,310 (98.18%)
3 large: 748,401 / 755,160 (99.10%)
4 large: 48,944 / 49,445 (98.99%)
Total: 10,924,808 / 11,905,457 (91.76%) or 8,961,082 / 9,343,307 (95.91%)
So not as good as you'd think, although its major downfall is 1L, and this can mostly be explained by 12 being a really rubbish large number on its own.
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Re: How many Numbers puzzles are there in total?
By the way, if you're only playing 4L, the best possible set of large numbers is 12, 14, 17, 57, with 49,438 / 49,445 (99.99%) of games solvable - only 7 games not solvable!
The unsolvable games are:
12, 14, 17, 57, 1, 1 ---> 344
12, 14, 17, 57, 1, 1 ---> 381
12, 14, 17, 57, 1, 1 ---> 423
12, 14, 17, 57, 1, 1 ---> 438
12, 14, 17, 57, 1, 1 ---> 471
12, 14, 17, 57, 1, 1 ---> 604
12, 14, 17, 57, 7, 7 ---> 622
The unsolvable games are:
12, 14, 17, 57, 1, 1 ---> 344
12, 14, 17, 57, 1, 1 ---> 381
12, 14, 17, 57, 1, 1 ---> 423
12, 14, 17, 57, 1, 1 ---> 438
12, 14, 17, 57, 1, 1 ---> 471
12, 14, 17, 57, 1, 1 ---> 604
12, 14, 17, 57, 7, 7 ---> 622
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Re: How many Numbers puzzles are there in total?
Which large numbers are generally most useful, across all combinations with other numbers, and which are least useful? 97 and 12 respectively?
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Re: How many Numbers puzzles are there in total?
Nope. The most useful numbers, in order (some omissions) are:Thomas Cappleman wrote:Which large numbers are generally most useful, across all combinations with other numbers, and which are least useful? 97 and 12 respectively?
11, 13, 12, 14, 17, ... , 92, 90, 98, 96, 100
97 comes in 15th from bottom. As you can see, 12 is 3rd. The numbers have been ranked by the statistic "Of all the possible numbers games which involve this number, how many are solvable?"
The reason for this apparent reversal is quite easy - 97 is really good at 1 large (it's the best number for it, in fact). It's not too bad at 2 large. It's quite a bad number for 3 or 4 large. With a fixed set of 4 numbers, there are far more possible 1 or 2 large games than 3 or 4 large games, so 97 comes out quite well.
On the other hand, smaller large numbers are quite good for 3 and 4 large (cf 12, 14, 17, 57 as above).
When we're allowed to have any large numbers we like, and haven't just fixed four of them in advance, there are WAY more 3 and 4 large games than there are 1 or 2 large, so the numbers that come out on top are those which are better for 3 or 4 large games, in general. That's why overall, small numbers make better larges - because overall counts skew 3 and 4 large in a dramatic way - but you still wouldn't have them in your dream team.
https://en.wikipedia.org/wiki/Simpson%27s_paradox
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Re: How many Numbers puzzles are there in total?
Which ones give the best average solvability across all sets of 4 (so for example 12 has 88.28% with 16, 18, 24, 91.76% with 37, 62, 87, etc. and then taking the average of these)? Was what I was originally thinking, though you've answered an equally interesting question anyway.Dave Ricesky wrote:Nope. The most useful numbers, in order (some omissions) are:Thomas Cappleman wrote:Which large numbers are generally most useful, across all combinations with other numbers, and which are least useful? 97 and 12 respectively?
11, 13, 12, 14, 17, ... , 92, 90, 98, 96, 100
97 comes in 15th from bottom. As you can see, 12 is 3rd. The numbers have been ranked by the statistic "Of all the possible numbers games which involve this number, how many are solvable?"
The reason for this apparent reversal is quite easy - 97 is really good at 1 large (it's the best number for it, in fact). It's not too bad at 2 large. It's quite a bad number for 3 or 4 large. With a fixed set of 4 numbers, there are far more possible 1 or 2 large games than 3 or 4 large games, so 97 comes out quite well.
On the other hand, smaller large numbers are quite good for 3 and 4 large (cf 12, 14, 17, 57 as above).
When we're allowed to have any large numbers we like, and haven't just fixed four of them in advance, there are WAY more 3 and 4 large games than there are 1 or 2 large, so the numbers that come out on top are those which are better for 3 or 4 large games, in general. That's why overall, small numbers make better larges - because overall counts skew 3 and 4 large in a dramatic way - but you still wouldn't have them in your dream team.
https://en.wikipedia.org/wiki/Simpson%27s_paradox
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Re: How many Numbers puzzles are there in total?
In that case, your hunch was spot-on:Thomas Cappleman wrote:Dave Ricesky wrote:Which ones give the best average solvability across all sets of 4 (so for example 12 has 88.28% with 16, 18, 24, 91.76% with 37, 62, 87, etc. and then taking the average of these)? Was what I was originally thinking, though you've answered an equally interesting question anyway.
97 - 92.277%
93 - 92.276%
83 - 92.255%
95 - 92.238%
87 - 92.233%
94 - 92.220%
...
11 - 91.544%
14 - 91.518%
18 - 91.469%
20 - 91.468%
16 - 91.444%
12 - 91.189%
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Re: How many Numbers puzzles are there in total?
Awesome work. I don't think this has been answered - what are the best 4 large numbers for each choice of number of large numbers? So for 4 large it's 12, 14, 17 and 57, but what about for 1, 2 or 3 large?
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Re: How many Numbers puzzles are there in total?
Please pardon the shamefully slow reply! I'm still a rather occasional lurker around these parts, and I stumbled across your post via the Impossible Rachel thread.Jon Corby wrote: Also, who is this (on the left) from the solver page that Jim linked to:
?
That is me on the right of shot, taking a bite out of a YTV biro. On the left is Ariane Sherine, a comedy writer, journalist and blogger. She's perhaps best known for spearheading the Atheist Bus Campaign.
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Re: How many Numbers puzzles are there in total?
I'm so glad that this thread exists.
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Re: How many Numbers puzzles are there in total?
Just realised I never replied to this, it's amazing. Probably one of my favourite posts ever. Thanks Dave!Dave Ricesky wrote:I have an answer for you, Charlie. The easiest four numbers are:Charlie Reams wrote:Great topic, enjoyed reading all this!
Harder question: if one were designing a new variant, which four large numbers (in the range 11-100) make the game hardest (i.e. fewest games solvable) and which easiest (i.e. most games solvable)?
*** Drum roll ***
59, 83, 93, 97, with 11,072,459 of the 11,905,457 games solvable, or 9,108,733 / 9,343,307 if you discount 6 small. That's 93.00% or 97.49% respectively.
The hardest four numbers are 12, 16, 18, 24, with 10,510,350 / 11,905,457 (88.28%) or 8,546,624 / 9,343,307 (91.47%).
The breakdowns by number of larges chosen is:
59, 83, 93, 97
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 5,063,270 / 5,221,392 (96.97%)
2 large: 3,259,190 / 3,317,310 (98.25%)
3 large: 738,513 / 755,160 (97.80%)
4 large: 47,760 / 49,445 (96.59%)
12, 16, 18, 24
--------------------------
6 small: 1,963,726 / 2,562,150 (76.64%)
1 large: 4,704,925/ 5,221,392 (90.11%)
2 large: 3,087,519 / 3,317,310 (93.07%)
3 large: 707,920 / 755,160 (93.74%)
4 large: 46,260 / 49,445 (93.56%)
25, 50, 75, 100 performs relatively poorly, coming 2,378,830th out of the 2,555,190 possible selections - that's in the 7th percentile.
It still beats 11, 12, 13, 14, which comes 2,554,989th out of 2,555,190, with 10,621,119 / 11,905,457 (89.21%) or 8,657,393 / 9,343,307 (92.66%)
97,98,99,100 is actually not too bad, coming 1,677,651th out of 2,555,190 (35th percentile), with counts of 10,933,717 / 11,905,457 (91.84%) or 8,969,991 / 9,343,307 (96.00%)