So, in the Fate mailing list, people are discussing what the various effects of replacing a Fudge die with a normal polyhedral die are. This is an interesting conversation to me, and I thought I'd do some math to get a grip on it...

4dF

First up is the standard 4dF distribution (which I've talked about before)

4dF | %
----|-------------
  -4|1.23456790123
  -3|4.93827160494
  -2|12.3456790123
  -1|19.7530864198
   0|23.4567901235
  +1|19.7530864198
  +2|12.3456790123
  +3|4.93827160494
  +4|1.23456790123

This is a baseline for my comparisons below.

3dF + 1d4

An interesting option is replacing 1dF by 1d4.

3dF + 1d4 | %
----------|--------------
        -2|0.925925925926
        -1|3.7037037037
         0|9.25925925926
        +1|15.7407407407
        +2|20.3703703704
        +3|20.3703703704
        +4|15.7407407407
        +5|9.25925925926
        +6|3.7037037037
        +7|0.925925925926

Here it's shifted the average from 0 to +2.5, and increased the range--but not uniformly! The low end has been shifted from -4 to -2, a +2 increase, while the high end has been shifted from +4 to +7, a +3 increase!

3dF + 1d6

A bit more extreme is replacing 1dF by 1d6.

3dF + 1d6 | %
----------|--------------
        -2|0.617283950617
        -1|2.46913580247
         0|6.17283950617
        +1|10.4938271605
        +2|14.1975308642
        +3|16.049382716
        +4|16.049382716
        +5|14.1975308642
        +6|10.4938271605
        +7|6.17283950617
        +8|2.46913580247
        +9|0.617283950617

The average has shifted from 0 to +3.5, a significant boost. Again, the low end has shifted by +2, but the high end has shifted from +4 to +9, a +5 shift! This seems very high-powered.

2dF + 2d4

A rather extreme option, this removes negative results from the equation altogether... though it does make a very nice bell curve centered around 5 in the range [0,10]. This could be very handy as a normalized d10 replacement, perhaps... but so could 4dF + 4, and that wouldn't have to worry about rerolling on 10. Anyways..

2dF + 2d4 | %
----------|--------------
         0|0.694444444444
        +1|2.77777777778
        +2|6.94444444444
        +3|12.5
        +4|17.3611111111
        +5|19.4444444444
        +6|17.3611111111
        +7|12.5
        +8|6.94444444444
        +9|2.77777777778
       +10|0.694444444444

Neat to me, perhaps not so much to others... If you're interested, here's an anydice link where you can see graphs and such, and play around with it yourself.

Thanks for reading!