# Solution to Random Hall

#### by Colin Lu, Azalea Weisblat, Jon Schneider, Robert Tunney, Anderson Wang

```HABITUAL
SOLE MEUNIERE
DRESS CODE
GIANT GROWTH
THE XX
PRODUCT TESTING```

In this puzzle, we are presented with a random number generator that generates sets of 17 numbers at a time. These numbers are, in fact, random. However, the key lies in determining what distributions they are generated from.

Here are a sample few rows of output from the generator:

```0.546  2.321  6.623  7.086  3.273  1.040  2.321  7.086  2.366  3.000  3.273  0.546  6.623  5.000  0.019  1.927  1.000
0.822  5.496  4.624  2.835  2.498  0.181  5.496  2.835  1.496  3.000  2.498  0.822  4.624  9.000  0.127  0.162  2.000
0.841  3.730  4.511  4.444  3.944  2.573  3.730  4.444  0.011  1.000  3.944  0.841  4.511  6.000  0.472  1.607  0.000
0.771  6.556  6.940  4.298  3.369  0.233  6.556  4.298  1.351  2.000  3.369  0.771  6.940  4.000  0.190  2.676  0.000
0.917  3.812  4.178  6.118  3.085  0.245  3.812  6.118  0.302  2.000  3.085  0.917  4.178  11.00  0.210  1.512  0.000
0.433  2.322  6.219  8.600  3.220  1.059  2.322  8.600  0.047  4.000  3.220  0.433  6.219  5.000  0.011  2.367  1.000
0.958  1.967  4.542  4.763  2.090  0.876  1.967  4.763  1.122  4.000  2.090  0.958  4.542  7.000  0.057  0.533  0.000
0.640  1.489  4.437  2.429  2.389  1.520  1.489  2.429  1.387  8.000  2.389  0.640  4.437  7.000  0.179  2.052  0.000
0.946  4.705  6.910  6.649  3.889  0.217  4.705  6.649  0.845  2.000  3.889  0.946  6.910  10.00  0.658  2.176  1.000
0.701  5.379  4.014  2.373  2.541  0.318  5.379  2.373  0.455  4.000  2.541  0.701  4.014  11.00  0.056  6.387  1.000
(A)    (B)    (C)    (D)    (E)    (F)    (G)    (H)    (I)    (J)    (K)    (L)    (M)    (N)    (O)    (P)    (Q)
```

For clarity, let's denote these columns A through Q. First, we note that there are several columns that are duplicates of each other: A and L, B and G, C and M, D and H, E and K.

Now we analyze these distributions. Some techniques we could use include making histograms, examining their mean/variance/median/excess kurtosis/skew, and noting properties like whether all outputs are integers or whether any are negative. Eventually we can determine that they each come from one of the following types of distributions:

• Normal
• Uniform
• Beta
• Exponential
• Chi Squared
• Poisson

At this point, we must return to the answers. Looking at them, we note that the answers correspond to the distributions. Normal is a synonym of `HABITUAL`; Poisson is French for fish, and `SOLE MEUNIERE` is a French fish dish. Chi looks like X, so chi squared corresponds to two Xes next to each other or `THE XX`. A `DRESS CODE` is like a uniform. `GIANT GROWTH` sounds like exponential growth. And `PRODUCT TESTING` is also known as beta testing.

Each distribution is defined by one or two parameters. For example, a normal distribution is defined by it's mean and variance. If we compute the parameters for each of our distributions we'll find that they are all positive integer values. Furthermore, the distributions with two parameters are also the distributions that are duplicated.

Distribution Answer Parameter 1 Parameter 2
Normal HABITUAL Mean (μ) Variance (σ2)
Uniform DRESS CODE Minimum (a) Maximum (b)
Beta PRODUCT TESTING α β
Exponential GIANT GROWTH Rate (λ)
Chi Squared THE XX Degrees of freedom (k)
Poisson SOLE MEUNIERE Rate (λ)

We can use the parameters as indexes into each distribution's corresponding answer. The first time we see a duplicated distribution, we should use the first parameter, and the second time we see it we should use the second parameter:

Distribution Param 1 Param 2 Answer Index 1 Index 2
(A) Beta 4 1 `PRODUCT TESTING` D P
(B) Normal 4 3 `HABITUAL` I B
(C) Uniform 4 7 `DRESS CODE` S O
(D) Normal 5 6 `HABITUAL` T U
(E) Uniform 2 4 `DRESS CODE` R S
(F) Exponential 2 `GIANT GROWTH` I
(G) Normal 4 3 `HABITUAL` I B
(H) Normal 5 6 `HABITUAL` T U
(I) Chi squared 1 `THE XX` T
(J) Poisson 4 `SOLE MEUNIERE` E
(K) Uniform 2 4 `DRESS CODE` R S
(L) Beta 4 1 `PRODUCT TESTING` D P
(M) Uniform 4 7 `DRESS CODE` S O
(N) Poisson 8 `SOLE MEUNIERE` N
(O) Exponential 6 `GIANT GROWTH` G
(P) Chi squared 3 `THE XX` E
(Q) Poisson 1 `SOLE MEUNIERE` S

If we take the indexed letters in the order of the columns (A through Q), we'll arrive at the answer, DISTRIBUTE SPONGES.