Pokemon is at its heart a game of mathematics and statistics, and as such offers many problems of surprising mathematical complexity. I have gathered three such problems, each of which is hefty enough that I will guarantee a badge and possibly more to anyone who solves even one correctly and writes it up. I will accept and publish onsite any papers offering rigorously derived exact solutions; approximative solutions will be reviewed on a case-by-case basis. If you find that you are only able to solve part of a problem, please do submit that paper anyway. There are no reservations, and there is no rush; even if a problem has been solved, I will accept as equal any solution that uses a substantively different method, or that is sufficiently simpler and more elegant than a previously accepted solution. I may occasionally add further problems as they occur to me. So have at it, and remember to thoroughly understand all relevant game mechanics before tackling a problem. If any part of any problem appears unclear, please point it out to me, and I will change it. Problem 1. Suppose that there is a Pokemon with the ability Moody on the field, and is kept on the field for an arbitrary number of turns. This Pokemon initially has no stat boosts or drops, and experiences no stat boosts or drops except by the effect of Moody. a) At the end of each turn, Moody will activate and change the Pokemon's stat distribution. For example, at the end of turn 1, the Pokemon will have +2 in one stat, -1 in another, and 0 in the five other stats. Calculate the probability distributions for an arbitrary stat's boost level (since no stat is unique, only one stat needs to be considered) at the end of turns 1 through 50. For example, the probability distribution for turn 1 is 1/7 for the -1 boost level, 1/7 for the +2 boost level, 5/7 for the 0 boost level, and 0 for any other boost level. b) Calculate the probability distributions for Stored Power's base power corresponding to the Pokemon's boost distribution at the end of turns 1 through 50. Problem 2. The first and second generations of Pokemon did not have EVs; instead, they had something similar, called stat experience. Since this is a less familiar topic, people considering this problem are strongly urged to read about the mechanics of stat experience before approaching it. a) The first part of this problem is a project that requires research on the first and second generation games. Suppose that six Pokemon participate in a battle against a Chansey. Do each of the Pokemon gain 1 stat experience point in Attack, or 0 stat experience points in Attack? b) Determine the minimum amount of Attack stat experience that may be earned while maximizing stat experience in every other stat in the following cases: i) A generation 1 Pokemon in a generation 1 game that cannot be traded to be trained in a generation 2 game ii) An arbitrary generation 1 Pokemon, with no special restrictions iii) A generation 2 Pokemon (obviously, cannot be traded to a generation 1 game) In all cases, do not consider any Pokemon that can only be battled by exploiting a glitch. Note that because of how the effects of stat experience are quantized, a stat is maximized once it reaches 63504 units of stat experience. For the same reason, instances like 14400 stat experience and 15375 stat experience are considered equivalent, because their effect is the same. c) Repeat part b for each of the other stats, namely HP, Defense, Speed, and Special (note that Special Attack and Special Defense are conflated). Problem 3. Consider a randomly generated EV spread that obeys the rules governing EV spreads. a) Taking into account the quantization of EV spreads (18 EVs and 16 EVs are equivalent, for example), calculate the number of distinct EV spreads possible in the following conditions. i) The EV spread totals to 510 EVs. ii) The EV spread may total to any positive integer not greater than 510 EVs. b)Calculate the probability distribution for EVs for a particular stat (that is, determine the probability that a given stat will have 0 EVs, 1 EV, 2 EVs, up through 255 EVs) in the following conditions: i) The EV spread totals to 510 EVs. ii) The EV spread may total to any positive integer not greater than 510 EVs. Note: Problem 3 derives substantially from a conversation I had with Antar about distributions of EV spreads in Challenge Cup.