I previously wrote on this blog that I am talking two courses this semester: one on scientific computing and one on planning algorithms. Coincidentally, I have recently played a couple games that offer nearly perfect examples to illustrate concepts that we are learning about in these courses. Here is the one about scientific computing the other one will come up soon.
Problem 13: Characteristic Characters
In the HERO System, player characters are defined in part by a list of numerical "characteristics," such as strength, intelligence, dexterity, defensive capability, etc. When creating characters, players spend "character points" (CP) to purchase characteristics. Each characteristic costs a different number of CP per point of the characteristic. Most characteristics only cost 1 CP per point, but some cost more. For example Dexterity costs 2 CP per point because it affects important combat factors such as initiative, and Speed costs a whopping 10 CP per point because every point of speed increases the number of "phases" you get to perform actions each turn. For example if I were to buy 15 points of Dexterity and 5 points of Speed, that would cost (15*2) + (5*10) = 70 total CP. Note that you ARE permitted to buy a negative amount of a Characteristic.
The rules for purchasing Characteristics changed significantly between the 5th edition and 6th edition of the game. In 5th edition, Characteristics were divided into two categories: "base characteristics" and "figured characteristics." Base characteristics were purchased as described above. However, "figured characteristics" each had a base value that you got for free, that was a linear function of the values of the base characteristics you purchased. For example, let's say "maximum hit points" had a base value of (3 * Constitution + 2 * Strength). Then if you bought 15 Constitution and 10 Strength, you would get (3*15 + 2*10) = 65 hit points "for free", and you could then buy more for a given cost in CP per point (or buy a negative amount, effectively "selling back" the hit points you got for free.)
In 6th edition, figured characteristics were eliminated; they instead became treated the same way as base characteristics - you don't get any "free points", you just have to buy them up as normal. Some of the CP-per-point costs of the various characteristics were altered to compensate.
Now here is the question:
Prove that regardless of what the costs and "figured characteristic functions" were in the old system, that it was possible to modify it - changing costs but eliminating figured characteristics - such that every possible combination of characteristics costs exactly the same under the new system as under the old system.
The solution is here.