The Special Fish Report
Albert W. Small (December 1944)
Codes and Ciphers
TOP SECRET Special Fish Report Page 24
I have already submitted as part of this report the famous
"Significance Test IV of the Black file." Patterns derived from the crude
convergence of the Delta-Z1, Delta-Z2 rectangle, for Delta-X1, are checked for
by means of this test. Actually the theta terms of the test are not computed,
the answers comes out close to the borderline.
It has been found that the theta terms seldom sum to more than 300,000/N.
Therefore the standard for the X1 wheel is taken as:
(2.17 X^2)/N > 219 - 300,000/N
or in simpler terms:
X > 10 (N-1500)^1/2
In the above, X = Sum |xi| wherein xi are the excesses of dots over crosses
in the resultant pattern.
From this formula in simplest terms is calculated a value for X
given each length N, which must be equalled or exceeded by the Delta-X1 wheel
before the rectangle is called significant.
The above formula is currently being "argued out;" some persons
favour substituting 1271 for the value 1500 claiming the latter too high;
others favour the older rule, X>(80N)^1/2. But the one given is the one
now in use, and is referred to in one of the screeds sent by Capt. Fried
as "the better rule." When the computors have converged a pattern to
this point they then turn the rectangle over to the wheel breakers.
A table showing results obtained with significant rectangles
follows. Jellyfish had not "come out" for six weeks, and Gurnard had "come out"
well. The table to follow compares the two.