What is the optimal method for getting every Unown in the Ruins of Alph?

Question

In the Ruins of Alph in the Johto region, you can introduce four groups of the Unown alphabet to the main chamber in whatever order you'd like. Each group is a different size.

If I wanted to capture every Unown letter as fast as possible, what is the optimal order for introducing and encountering Unown, and why?

Answer 1

Introduce every group, then start catching.

Learned myself a new mathematical concept.

This question is a variation on the Coupon Collector's problem: "Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least once?" the simplified solution being:

E(T) = n * H_{n - k}

Where E(T) is the estimated number of draws required, n is the number of unique coupons there are, k is the number of unique coupons you've already drawn before starting the trial, and H_{n - k} being the (n - k)th harmonic number.

So, if we introduce the smallest group of Unown first, catch them all, then introduce the next smallest and so forth, the expected number of encounters looks like this:

E(T) = 3 * H₃ + 8 * H₅ + 15 * H₇ + 26 * H₁₁ = 141.176

Contrasting introducing the largest group first and going smaller and smaller, we get:

E(T) = 11 * H₁₁ + 18 * H₇ + 23 * H₅ + 26 * H₃ = 180.073

Finally, let's look at releasing all the Unown at once and then starting the hunt:

E(T) = 26 * H₂₆ = 100.215

So as it would turn out, releasing all the Unown at once is far and away the best solution.

I wasn't factoring in Unown Z's reduced spawn rate in the calculation, so it may be worthwhile (for peace of mind if nothing else) to get X, Y and Z first anyway.

Answer 2

It could be easier in some cases to complete the Ho-oh puzzle first. While it may be the slightly statistically worse option, completing that puzzle releases Unown X,Y and Z, and Unown Z has a lower chance to appear due to the way the Unown form is determined in an encounter.

It would therefore make sense to hunt for Z when you have less Unown released, giving you a greater chance of encountering it.

Incidentally, here is the list of each Unown released when a puzzle is completed:

• Kabuto - unlocks 11 letters - A, B, C, D, E, F, G, H, I, J, K
• Omanyte - unlocks 7 letters - L, M, N, O, P, Q, R
• Aerodactyl - unlocks 5 letters - S, T, U, V, W
• Ho-oh - unlocks 3 letters - X, Y, Z