'Skipjack' dice resolution

In role playing games, you use the dice to determine success or failure, and degrees thereof.  In modern d20 systems, you roll a 20 sided die (d20), add bonuses, and see if you meet or exceed a target number.  Rolling a 1 always fails, rolling a 20 always succeeds.

I don't like having to do math every time I roll the die.

There is a popular alternative method - roll under.  With this, you use some target number (normally based on your character's attributes), roll a d20, and try to get less than or equal to the number.  Very simple.  But if you want to have a variable difficulty, you still need to add or subtract from your target number.

I don't like having to do math every time I roll the die.

A closely related mechanic is called blackjack resolution (or under-over).  You need to roll equal to or less than your target number, but higher than the difficulty.  You don't have to do math when you roll, you just look at the die and compare it to the two numbers.

Now we're getting somewhere!

I like blackjack resolution, but I also like to make everything possible.  If the difficulty is too high, the player will be unable to succeed - and then what's the point of rolling the die?  I also like to keep everybody involved.  So, I use what I call skipjack.  (What can I say, I like tuna.)  It's like blackjack, but instead of a fixed difficulty number, you roll a difficulty die.  The greater the difficulty, the larger the die.  (0, d4, d6, d8, d10, d12, d20)  Now the player rolls a d20, and the game master rolls the difficulty die (or vice versa).  The d20 still has to show less than or equal to the player character's appropriate attribute, but higher than the difficulty die result.  The difficulty can vary, but nothing is impossible.

Players rolling to defend themselves feels really good.

This opens up a few opportunities.  For example, magic armor could us a larger die size, or allow a roll with advantage (roll twice, take the better result).  Stacked defenses/difficulties could grow the die size, or just roll separate dice.  Character level could determine their die size when defending.

Note:  It's important, when using a roll-under mechanic, to start your characters out using a more limited attribute range than the traditional 3d6 (3 to 18).  That method leads to way too much variability in starting power levels.  Instead, I use 2d3+6, giving a starting range of 8 to 12.  This is identical to a starting bonus of -2 to +2, which falls in line with the old D&D standards.  I also allow characters to add half their level to their attribute, when doing something related to their class (profession) or background.  This adjusted number can be written on their character sheet, so no math is involved during the game.