How many Numbers puzzles are there in total?

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Ian Volante
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Re: How many Numbers puzzles are there in total?

Post by Ian Volante » Sat Jan 23, 2016 10:16 am

Dave Ricesky wrote:
Ian Volante wrote:
Dave Ricesky wrote:you can reach values larger than 2^31
Really? How, given that 25*50*75*100*10*10 is under 1x10^9 and 2^31 is >2x10^9? Some intermediate factoring steps?
Charlie's question was about using different sets of large numbers - and 100*99*98*97*10*10 is way too big, for example.
Oh yes. *winds neck in*
meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles meles

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Fri Jan 29, 2016 11:32 am

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)?
I have an answer for you, Charlie. The easiest four numbers are:

*** 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?

Post by Johnny Canuck » Fri Jan 29, 2016 2:57 pm

Dave Ricesky wrote:
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)?
I have an answer for you, Charlie. The easiest four numbers are:

*** 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%)
Where do 12, 37, 62, 87 fall?
Otherwise known as "Will", "JC", or "that guy who's bad at The Mole".

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Re: How many Numbers puzzles are there in total?

Post by Thomas Cappleman » Fri Jan 29, 2016 3:17 pm

Johnny Canuck wrote:Where do 12, 37, 62, 87 fall?
2 prime numbers, and 62 having 31 (a prime greater than 10) as a factor should put it much higher than the default.

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Re: How many Numbers puzzles are there in total?

Post by Clive Brooker » Fri Jan 29, 2016 3:19 pm

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.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Fri Jan 29, 2016 3:34 pm

Thomas Cappleman wrote:
Johnny Canuck wrote:Where do 12, 37, 62, 87 fall?
2 prime numbers, and 62 having 31 (a prime greater than 10) as a factor should put it much higher than the default.
87 = 3*29, but anyway...

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?

Post by Dave Ricesky » Fri Jan 29, 2016 3:47 pm

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

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Re: How many Numbers puzzles are there in total?

Post by Thomas Cappleman » Fri Jan 29, 2016 3:48 pm

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?

Post by Dave Ricesky » Fri Jan 29, 2016 4:04 pm

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?
Nope. The most useful numbers, in order (some omissions) are:

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?

Post by Thomas Cappleman » Fri Jan 29, 2016 4:13 pm

Dave Ricesky wrote:
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?
Nope. The most useful numbers, in order (some omissions) are:

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
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.

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Re: How many Numbers puzzles are there in total?

Post by Dave Ricesky » Fri Jan 29, 2016 4:46 pm

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.
In that case, your hunch was spot-on:


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?

Post by Gavin Chipper » Sun Jan 31, 2016 6:19 pm

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?

Post by Chris Philpot » Sat Feb 20, 2016 8:49 pm

Jon Corby wrote: Also, who is this (on the left) from the solver page that Jim linked to:

Image

?
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.

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?

Post by Matty Artell » Sat Feb 20, 2016 9:16 pm

I'm so glad that this thread exists.

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Re: How many Numbers puzzles are there in total?

Post by Charlie Reams » Sat Dec 17, 2016 10:12 am

Dave Ricesky wrote:
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)?
I have an answer for you, Charlie. The easiest four numbers are:

*** 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%)
Just realised I never replied to this, it's amazing. Probably one of my favourite posts ever. Thanks Dave!

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