the Savage Worlds (SWADE edition) base mechanics are quite simple:
[ +- ] SWADE basic rules
-Attributes and Skills are ranked by increasing die type: d4, d6, d8, d10, d12 (the higher the die the better you are at that skill).
For example Alice starts with d6 in Athletics. If you are Unskilled at something, you roll 1d4-2
-all dice rolls in SW can ace: this is when you roll the max possible number on a die. A d4 aces on a 4, a d8 aces on an 8, etc. When this happens, you reroll the same die and add it to the total.
Example: I roll an Athletics d6: first die gives me a 6, an ace. I reroll and get a 3. My final total is 6+3=9
-characters (and some villains/NPCs too) are Wild Cards, more heroic than average. This means that in addition to your skill die, you also roll a Wild Die, an extra d6. This can ace as usual. You choose the highest die as your final result.
For example, to roll Athletics, Alice would roll [1d6 Athletics] and [1d6 Wild] and choose the highest.
The Wild Die for an Unskilled check is 1d6-2. So if I am unskilled at Research, I would roll: 1d4-2, 1d6-2 and choose the highest.
-the target number for a regular skill check is a 4. This can be modified for difficulty.
-the target number to hit someone is the target's PARRY stat.
-the target number to inflict damage after hitting is the target's TOUGHNESS stat.
-the target number in an opposed roll is whatever the opponent rolls.
-levels of success: if you roll equal to the target number you succeed, and each increment of 4 above that success is called a raise, giving extra bonuses.
For example, Alice is climbing a regular wall and the target number is 4. She rolls Athletics (d6) and Wild die (d6) and gets [6+6+2 acing twice =14] , [5]. She chooses the highest (14).
The result is a roll of 14 vs target 4: 4s 8r1 12r2 (she has a success reaching 4, one raise reaching 8 and a second raise reaching 12. The excess (13, 14) are lost because they do not reach the next raise (raises are every increment of 4).
In this case, Alice climbs with great ease, and I would give her extra advantages, for example she might land on the other side much ahead of where she would normally land, or catch someone she is pursuing, etc.
