[dake50]

I have been charting tricks won vs hcp and find 22 succeeds 1/3 as 21 and 1/2 as 23.

Finding no reason to explain this, are others finding this anomaly? Is 22 just a confluence of overrated Q/J for example? Is 22 "heavy" in quacks?

[nigel_k]

We need more information.

1. Tricks won under what conditions?
2. What do you mean by 'succeeds 1/3 as 21 and 1/2 as 23'

[TylerE]

I strongly suspect insufficient sample size.

[MFA]

dake50, on Sep 20 2009, 08:56 PM, said:

I have been charting tricks won vs hcp and find 22 succeeds 1/3 as 21 and 1/2 as 23.
...

No idea what this means.

If it means that a 22 hcp hand takes fewer tricks than a 21 hcp hand, then you have just proven that there is a bug in your program.

[dake50]

No one else seeing an anomaly at 22, I take it?? Let alone at 7 and 12 but lesser?

[phil_20686]

Sample size is one obvious possibility, however it strikes me that there is a deeper reason why such an anomaly *might* exist. It's well known that the milton point count does not accurately represent the trick taking ability of the high cards. In particular, aces are worth much more than 4. There could be some subtle combinatorics here that suggest that some numbers are much more likely to include larger numbers of queens and Jacks than aces and kings.

Let us take 4 posints as a simple example,

there are 4 ways to make this with aces,
there are 6 ways to make thsi with two queens
there is 1 way to make this with 4 jacks
there are 16 ways to make this with a K and a J.
There are 24 ways to make this with a Q and two jacks.
51 total

Average nuimbers:

0.08 aces
0.31 kings
0.71 queens
1.33 jacks

Thus the average here is skewed hugely in favour of soft values, let us conisder a neighbouring example


Let us take say 5 points
there are 16 ways to make this with an ace and a J
16 ways to make this with a K and Q
there are 24 ways to make this with a K and two jacks
there are 24 ways to make this with two Q's and a jack
there are 16 ways to make this with a Q and 3 jacks.
86 hands total
Average number
0.19
0.47
0.93
1.77

Thus adding a single point more than doubles the expected number of aces but has a smaller effect on kings. I beleive this to be a feature of combinatorics with a limited number of things - there will be 'magic numbers' that maximise the chance of aces per HCP. Put another way, if i used a more complex ways of measuring the value of aces, that mroe closely tied their value to their trick taking potential, say the zar perscription, then i might find that the cominatorics means that my value scores to not vary smoothly with my milton points.

here say my zar HCP points, 6,4,2,1 gives an average 4.47 and 6.65, which is a lto more than the 1.3 increase you might expect from changing from a 10 to a 13 point scale. Thus my value per Milton is more on the 5 point hand than on the 4 point hand. With numbers close to 20 one might expect the largest variation, as that should be where the cominatorics should maximise the aces per HCP (i guestimate).

EDIT: i hope niobody read this before I realised i had cacluated the permutations not the cominations

PS I do not have anything like the patience to work out the cominatorics for 21 22 and 23 HCP.

[hanp]

The strange properties of the number 22 have long been discussed by mathematicians and devil worshippers alike. For example, it is well known (but has never been proved!) among number theorists that of all the even numbers, 22 most closely resembles a prime number. And the double double (2 times 11) has been given satanic properties in many obscure cults.

I am not surprised that you find anomalies for hands with 22 HCP.

[matmat]

hanp, on Sep 22 2009, 11:37 AM, said:

The strange properties of the number 22 have long been discussed by mathematicians and devil worshippers alike. For example, it is well known (but has never been proved!) among number theorists that of all the even numbers, 22 most closely resembles a prime number. And the double double (2 times 11) has been given satanic properties in many obscure cults.

I am not surprised that you find anomalies for hands with 22 HCP.

it's also the number of players that start on the pitch of a football match.

[kenrexford]

22 is also the average age for graduating college. I'm not sure what this means, but it seems to be relevant. In college, especially toward the end, the number of tricks available seems high. Right after you graduate, however, the number seems to fall precipitously for that first year out, until it starts increasing again.

[Cascade]

hanp, on Sep 23 2009, 04:37 AM, said:

it is well known (but has never been proved!) among number theorists that of all the even numbers, 22 most closely resembles a prime number.

It is easily disproved.

Consider 2.

QED

[Jlall]

I am 22 and obv make more tricks than the 23 year olds!

[Cascade]

I used to be 22 and obv make more tricks than those who used to be 23 year olds!

Hang on that doesn't work.

[gwnn]

and I'm 22 and I take fewer tricks than 21 year olds

[maggieb]

Jlall, on Sep 22 2009, 03:22 PM, said:

I am 22 and obv make more tricks than the 23 year olds!

That makes you an anomaly.

[helene_t]

The OP is gibberish to me, but if it means that 22 takes fewer tricks than 21 or more than 23, then it must be a sample size issue.

If it means something more subtle, say that 22 vs (21+23)/2 break an otherwise apparent convex (or concave?) trend, then it might be interesting.

[kenrexford]

maggieb, on Sep 22 2009, 04:08 PM, said:

Jlall, on Sep 22 2009, 03:22 PM, said:

I am 22 and obv make more tricks than the 23 year olds!

That makes you an anomaly.

When he reaches 23, then he can let us know. Until then, he's guessing.

[Mbodell]

phil_20686, on Sep 22 2009, 08:32 AM, said:

Sample size is one obvious possibility, however it strikes me that there is a deeper reason why such an anomaly *might* exist. It's well known that the milton point count does not accurately represent the trick taking ability of the high cards. In particular, aces are worth much more than 4. There could be some subtle combinatorics here that suggest that some numbers are much more likely to include larger numbers of queens and Jacks than aces and kings.

PS I do not have anything like the patience to work out the cominatorics for 21 22 and 23 HCP.

I get that there are 4380 different combinations of HCP to get 22 and that we expect on average:

2.276 A
2.195 K
2.127 Q
2.059 J

Which, if anything, seems to favor A over the others. But when I look at 23 HCP I get out of the 4064 possibilities an average of:

2.413 A
2.292 K
2.189 Q
2.092 J

and we can see that the extra point is spread 0.138 A, 0.098 K, 0.062 Q, 0.032 J. This is not surprising that it takes a lot more tricks.

But we want to see 21 HCP take more tricks too. There are 4580 possible combinations of HCP to get 21 and we expect on average:

2.141 A
2.096 K
2.058 Q
2.031 J

and we can see that going from 21 to 22 adds the extra HCP through 0.134 A, 0.098 K, 0.069 Q, 0.029 J which is quite similar to the extra HCP addition we had going from 22 to 23.

So we can see that each point around here does increase the expected Aces more than it increases the expected J's, however, there is no reason to suspect that 22 is some anomaly in the trick taking distributions.

[Mbodell]

Hmm, I just realized the mistake that phil and I were making is we were only considering the number of combinations of point cards while not considering the non-point cards. There are 6 ways to make 22 points with only A and K, but these ways use only 6 cards and leave 8347680 ways to fill in the hand with cards 2-T for 50086080 total such hands. Contrast with AKKQQQQJJJJ which gives 24 ways to make the honors but is *not* 4 times more likely than the AAAAKK hands because it takes 11 cards and thus there are only 630 ways to fill the hand with cards 2-T and 15120 total such hands.

My revised calculations give for 21,22,23 HCP:

21: 2.695 A 2.073 K 1.488 Q 1.023 J
22: 2.846 A 2.167 K 1.549 Q 1.014 J
23: 2.985 A 2.290 K 1.597 Q 0.998 J

So again the expected number of Jacks is actually going down slightly, and the number of AKQ are all going up. And A's are even more common than what I wrote before (which makes sense because they give the most points per card leaving the most spaces for 2-T fillers). But it doesn't look like 22 is special in any way.

[Cascade]

Expected aces, kings, queens, jacks and probably tens are surprisingly linear from around 5 HCP to 30 HCP.

My rounded rules of thumb are:

ACES = 0.15 * HCP - 0.5

KINGS = 0.1 * HCP

QUEENS = 0.05 * HCP + 0.5

JACKS = 1

The correlation for these (when unrounded) is around the high 90s.

So each HCP extra on average produces about 0.15 extra ace, 0.1 extra king, 0.05 extra queen and no extra jacks. And this is fairly independent of the number of HCP.

[Cascade]

Here are the unrounded (well actually just more accurately rounded straight out of my spreadsheet) numbers:

ACES = 0.15080137869129 * HCP - 0.50274312636775

KINGS = 0.099957885163922 * HCP - 0.000661360065993

QUEENS = 0.046115866115133 * HCP + 0.54936553633723

JACKS = -0.000338630416366 * HCP + 1.01687231334264

I am not 100% sure over what range of HCP these regressions were calculated. I can check when I get home if anyone is interested but I am away from home for about 10 days.