GURPS lets players buy skills for their characters at different costs depending on the levels of the characters' attributes and the difficulty of each skill. I will focus on the IQ attribute and its related skills for now. IQ is purchased with points, ranges from 1 to infinity (practically 20), and defaults to an average score of 10. Increasing IQ costs 10 points for each of the first three increases above 10, then 15 points for each of the next two steps, 20 points for each of the next two steps, and 25 points for each subsequent step. So, having an IQ of 18 costs 125 points. A character can get points back by having a below average IQ, earning 10 points for each level below 10 (except, inexplicably, the only 5 points for level 8).
Skills are either easy, average, hard, or very hard. There are very many mental skills. For 1 point, you can get an easy skill at the same level as your IQ, an average skill at IQ-1, a hard skill at IQ-2, or a very hard skill at IQ-3. Spending half a point instead of 1 reduces the skill level by one (this is the lowest you can go), or you can increase your skill level by 1 for an additional 1 point (additional increases cost more per point: 2 each for the next three increases, then 4 each above that).
To succeed at a task, you roll 3d6 (3 six-sided dice) and sum the results. If the sum is less than or equal to your skill level, you succeed, but a sum of 17 or 18 always fails (critical successes and failures are not important for this analysis). Given our knowledge of the distribution of results on 3d6, we can figure out the best way to spend our points on IQ and skills for our characters.
Step 1) Figure out what mental skills you want your character to have, and what percent of attempts you want your character to be successful using them.
Step 2) Consult the table:
This shows the optimal IQ to have for a given number of average difficulty level skills (or their equivalents, see notes below) at a given average success rate (I did not bother showing success rates below 25%).
*Though an IQ of 3 is technically optimal in these cells, a 9 may be a better choice because the point cost difference is very small and IQ will have impacts on game play beyond the cost of skills.
How to convert non-average difficulty skills into average skill equivalents:
- White cells: Easy skills count as .5, Hard skills count as 2, Very Hard skills count as 3.
- Black cells: Easy skills count as .5, Hard skills count as 1.5, Very Hard skills count as 2.
- Medium grey cells with white text: Easy count as .67; Hard count as 1.33, Very Hard count as 2.
- Light grey cells: Easy count as .75, Hard count as 1.25, Very Hard count as 2.
- Medium grey cells with black text: conversion is variable and not worth including
Derp is a big tough guy who lets his sword do most of the thinking for him. He doesn't need many mental skills at all. He takes four, but he wants to be really good at them because failure makes him mad. An IQ of 3 would let him be regularly successful with his four mental skills while freeing up the most points to spend elsewhere, like on ST, DX, and HT. An IQ of 9 may be a small but worthwhile expense, though, depending on role-playing goals.
Brainiac McGee knows how to do everything. He got his first doctorate at the age of 7, and went back for more. He is never (hardly ever) wrong, so the desire success rate for everything is 98.1%. Since he'll have more than 9 mental skills, he'll definitely be in the white cell area, so very hard skills count as 3 average skills, and hard skills count as 2. Somewhere between 40 and 50 average skill equivalents is where an IQ of 18 is a better buy than an IQ of 17.
Bob is a suave dude. He's got a good mix of social and professional skills. He's not the best, but he gets by pretty well. With a success rate target in the mid 80%s, an IQ of 13 is best for 5-14 average skills. If he wants to be better at what he does, and has more than 7 skills, he should increase his IQ. If he has 7 or fewer skills and wants to be better, he should just spend points on the skills. An IQ of 13 is great for someone who wants to excel at a few core mental skills, especially if they are hard.
If you want to have a bunch of skills at different success rates, you're on your own. This table was a beast, and I'm not making one for each of the thousands of likely combinations of skill success rates that people might want. I recommend just planning to be really good at whatever skills you want, and working from there. Or just have an IQ of 13 and don't think about it too much.