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The Sum of the Squares of the Suit Lengths
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<h1> The Sum of the Squares of the Suit Lengths</h1>
<h2>A Measure of the "Sickness" of a Hand Pattern </h2>
In any hand, the average of the suit lengths is 3.25 = 13/4, obviously,
but what is the standard deviation of the suit lengths?
<p>
The standard deviation, it turns out, is related to
the sum of the squares of the suit lengths. In particular, the higher
the sum of the squares, the higher the standard deviation.
<p>
So it should come as no surprise that the more balanced hands should
have lower standard deviations, and hence lower sums of squares. In
particular, when the sum of the squares of the suit lengths is less
than or equal to 47, the hand is "balanced" in the traditional sense:
No singleton or void, and at most one doubleton.
<p>
Since the smallest value is 43, and all values are odd, we can "normalize"
this metric by subtracting 43 and dividing by 2.
<p>
We get the following table:
<pre class="codesample" style="margin-left:0">
Normalized | Squares sum | Std Dev | Shapes
============================================
0 | 43 | 0.5000 | 4-3-3-3
1 | 45 | 0.9574 | 4-4-3-2
2 | 47 | 1.2583 | 5-3-3-2
3 | 49 | 1.5000 | 4-4-4-1, 5-4-2-2
4 | 51 | 1.7078 | 5-4-3-1
5 | 53 | 1.8930 | 6-3-2-2
6 | 55 | 2.0616 | 5-5-2-1,6-3-3-1
7 | 57 | 2.2166 | 5-4-4-0,6-4-2-1
8 | 59 | 2.3629 | 5-5-3-0
9 | 61 | 2.5000 | 7-2-2-2,6-4-3-0
10 | 63 | 2.6300 | 7-3-2-1,6-5-1-1
11 | 65 | 2.7538 | 6-5-2-0
12 | 67 | 2.8723 | 7-3-3-0, 7-4-1-1
13 | 69 | 2.9860 | 7-4-2-0
15 | 73 | 3.2016 | 6-6-1-0, 8-2-2-1
16 | 75 | 3.3040 | 7-5-1-0, 8-3-1-1
17 | 77 | 3.4034 | 8-3-2-0
19 | 81 | 3.5940 | 8-4-1-0
21 | 85 | 3.7749 | 7-6-0-0
22 | 87 | 3.8622 | 9-2-1-1
23 | 89 | 3.9476 | 8-5-0-0, 9-2-2-0
24 | 91 | 4.0311 | 9-3-1-0
27 | 97 | 4.2720 | 9-4-0-0
30 | 103 | 4.5000 | 10-1-1-1
31 | 105 | 4.5735 | 10-2-1-0
33 | 109 | 4.7170 | 10-3-0-0
40 | 123 | 5.1881 | 11-1-1-0
41 | 125 | 5.2519 | 11-2-0-0
51 | 145 | 5.8524 | 12-1-0-0
63 | 169 | 6.5000 | 13-0-0-0
</pre>
In addition, one can get the wildness of a deal by adding up the sum of the
squares of all 16 suit lengths, or add up the normalized values. Again,
this is correlated to the standard deviation of the suit lengths.
<p>
We can also measure the wildness of a "fit" by summing the squares of the fits
in each suit. So if I'm 5-3-3-2 and partner is 4-3-2-4, our "fit" is
9-6-5-6. One thing interesting about this is that our opponent's fit
wildness is the same as our fit pattern wildness. That's because if our
pattern is: s-h-d-c, their pattern is (13-s)-(13-h)-(13-d)-(13-c), and the
sum of the squares of these values is:
<pre class="codesample" style="margin-left:0">
(13-s)^s+(13-h)^s+(13-d)^2+(13-c)^2=
s^2 + h^2 + d^2 + c^2 - 26*(s+h+d+c) + 4*13^2
</pre>
But (s+h+d+c) is 26, so the last two terms cancel, and we are left with the
original value.
<p>
Of course, if you think in terms of standard deviation, this makes more sense
than the pure calculation - the opponents' pattern has the same relative
distribution, just inverted, so the deviation should be the same.
<p>
This table can also be normalized, by subtracting 170 and dividing by 2.
There are 103 fit patterns, or 65 if we consider our fit pattern and opponent's
as the same (e.g., that 8-6-6-6 is the same as 7-7-7-5. If the sum of the longest fit and the shortest fit is 13, then the pattern is self-dual, for example, if our fit pattern is 9-7-6-4, then so is the opponent's.)
<pre class="codesample" style="margin-left:0">
Normalized | Squares sum | Patterns
============================================
0 | 170 | 7-7-6-6
1 | 172 | 8-6-6-6,7-7-7-5
2 | 174 | 8-7-6-5
4 | 178 | 9-6-6-5,8-8-5-5,8-7-7-4
5 | 180 | 9-7-5-5,8-8-6-4
6 | 182 | 9-7-6-4
8 | 186 | 10-6-5-5,9-8-5-4,8-8-7-3
9 | 188 | 10-6-6-4,9-7-7-3
10 | 190 | 10-7-5-4,9-8-6-3
12 | 194 | 10-7-6-3,9-9-4-4
13 | 196 | 11-5-5-5,10-8-4-4,9-9-5-3,8-8-8-2
14 | 198 | 11-6-5-4,10-8-5-3,9-8-7-2
16 | 202 | 11-7-4-4,11-6-6-3,10-7-7-2,9-9-6-2
17 | 204 | 11-7-5-3,10-8-6-2
18 | 206 | 10-9-4-3
20 | 210 | 12-5-5-4,11-8-4-3,11-7-6-2,10-9-5-2,9-8-8-1
21 | 212 | 12-6-4-4,9-9-7-1
22 | 214 | 12-6-5-3,11-8-5-2,10-8-7-1
24 | 218 | 12-7-4-3,10-10-3-3,10-9-6-1
25 | 220 | 12-6-6-2,11-9-3-3,11-7-7-1,10-10-4-2
26 | 222 | 12-7-5-2,11-9-4-2,11-8-6-1
28 | 226 | 13-5-4-4,12-8-3-3,10-10-5-1,9-9-8-0
29 | 228 | 13-5-5-3,12-8-4-2,11-9-5-1,10-8-8-0
30 | 230 | 13-6-4-3,12-7-6-1,10-9-7-0
32 | 234 | 13-6-5-2,12-8-5-1,11-10-3-2,11-8-7-0
33 | 236 | 13-7-3-3,10-10-6-0
34 | 238 | 13-7-4-2,12-9-3-2,11-10-4-1,11-9-6-0
36 | 242 | 13-6-6-1,12-9-4-1,12-7-7-0
37 | 244 | 13-7-5-1,12-8-6-0
38 | 246 | 13-8-3-2,11-10-5-0
40 | 250 | 13-8-4-1,12-9-5-0,11-11-2-2
41 | 252 | 12-10-2-2,11-11-3-1
42 | 254 | 13-7-6-0,12-10-3-1
44 | 258 | 13-9-2-2,13-8-5-0,11-11-4-0
45 | 260 | 13-9-3-1,12-10-4-0
48 | 266 | 13-9-4-0
50 | 270 | 12-11-2-1
52 | 274 | 13-10-2-1,12-11-3-0
54 | 278 | 13-10-3-0
60 | 290 | 12-12-1-1
61 | 292 | 13-11-1-1,12-12-2-0
62 | 294 | 13-11-2-0
72 | 314 | 13-12-1-0
84 | 338 | 13-13-0-0
</pre>
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