These graphs and tables show the bell curves for various numbers of dice vs. drawing the same number of cards from the 20-card deck. Most of these card bell curves match the d6 bell curve eerily well.

1d6 vs. 1-Card Bell Curve

Roll1d6
Freq
1‑Card
Freq
 1d6
%
1‑Card
%
 1d6
Cum%
1‑Card
Cum%
0010.00%5.00%0.00%5.00%
11216.67%10.00%16.67%15.00%
21316.67%15.00%33.33%30.00%
31416.67%20.00%50.00%50.00%
41416.67%20.00%66.67%70.00%
51316.67%15.00%83.33%85.00%
61216.67%10.00%100.00%95.00%
7010.00%5.00%100.00%100.00%
Total620

 

2d6 vs. 2‑Cards Bell Curve

Roll2d6
Freq
2‑Card
Freq
 2d6
%
2‑Card
%
 2d6
Cum%
2‑Card
Cum%
1040.00%1.05%0.00%1.05%
2182.78%2.11%2.78%3.16%
32205.56%5.26%8.33%8.42%
43308.33%7.89%16.67%16.32%
544611.11%12.11%27.78%28.42%
655213.89%13.68%41.67%42.11%
766016.67%15.79%58.33%57.89%
855213.89%13.68%72.22%71.58%
944611.11%12.11%83.33%83.68%
103308.33%7.89%91.67%91.58%
112205.56%5.26%97.22%96.84%
12182.78%2.11%100.00%98.95%
13040.00%1.05%100.00%100.00%
Total36380

 

3d6 vs. 3‑Cards Bell Curve

Roll3d6
Freq
3‑Card
Freq
 3d6
%
3‑Card
%
 3d6
Cum%
3‑Card
Cum%
2060.00%0.09%0.00%0.09%
31360.46%0.53%0.46%0.61%
43841.39%1.23%1.85%1.84%
561802.78%2.63%4.63%4.47%
6103184.63%4.65%9.26%9.12%
7154806.94%7.02%16.20%16.14%
8216489.72%9.47%25.93%25.61%
92579811.57%11.67%37.50%37.28%
102787012.50%12.72%50.00%50.00%
112787012.50%12.72%62.50%62.72%
122579811.57%11.67%74.07%74.39%
13216489.72%9.47%83.80%83.86%
14154806.94%7.02%90.74%90.88%
15103184.63%4.65%95.37%95.53%
1661802.78%2.63%98.15%98.16%
173841.39%1.23%99.54%99.39%
181360.46%0.53%100.00%99.91%
19060.00%0.09%100.00%100.00%
Total2166,840

 

4d6 vs. 4‑Cards Bell Curve

Roll4d6
Freq
4‑Card
Freq
 4d6
%
4‑Card
%
 4d6
Cum%
4‑Card
Cum%
41720.08%0.06%0.08%0.06%
542400.31%0.21%0.39%0.27%
6107680.77%0.66%1.16%0.93%
7201,5601.54%1.34%2.70%2.27%
8352,9762.70%2.56%5.40%4.83%
9564,7764.32%4.11%9.72%8.94%
10807,1526.17%6.15%15.90%15.09%
111049,3368.02%8.03%23.92%23.12%
1212511,5929.65%9.97%33.56%33.09%
1314012,88810.80%11.08%44.37%44.17%
1414613,56011.27%11.66%55.63%55.83%
1514012,88810.80%11.08%66.44%66.91%
1612511,5929.65%9.97%76.08%76.88%
171049,3368.02%8.03%84.10%84.91%
18807,1526.17%6.15%90.28%91.06%
19564,7764.32%4.11%94.60%95.17%
20352,9762.70%2.56%97.30%97.73%
21201,5601.54%1.34%98.84%99.07%
22107680.77%0.66%99.61%99.73%
2342400.31%0.21%99.92%99.94%
241720.08%0.06%100.00%100.00%
Total1,296116,280

 

5d6 vs. 5‑Cards Bell Curve

Roll5d6
Freq
5‑Card
Freq
 5d6
%
5‑Card
%
 5d6
Cum%
5‑Card
Cum%
5100.01%0.00%0.01%0.00%
653600.06%0.02%0.08%0.02%
7151,6800.19%0.09%0.27%0.11%
8355,1600.45%0.28%0.72%0.39%
97012,1200.90%0.65%1.62%1.04%
1012624,7201.62%1.33%3.24%2.37%
1120543,0802.64%2.32%5.88%4.68%
1230568,4003.92%3.68%9.80%8.36%
1342098,6405.40%5.30%15.20%13.66%
14540131,1606.94%7.05%22.15%20.71%
15651161,5208.37%8.68%30.52%29.39%
16735185,2809.45%9.96%39.97%39.35%
17780198,12010.03%10.65%50.00%50.00%
18780198,12010.03%10.65%60.03%60.65%
19735185,2809.45%9.96%69.48%70.61%
20651161,5208.37%8.68%77.85%79.29%
21540131,1606.94%7.05%84.80%86.34%
2242098,6405.40%5.30%90.20%91.64%
2330568,4003.92%3.68%94.12%95.32%
2420543,0802.64%2.32%96.76%97.63%
2512624,7201.62%1.33%98.38%98.96%
267012,1200.90%0.65%99.28%99.61%
27355,1600.45%0.28%99.73%99.89%
28151,6800.19%0.09%99.92%99.98%
2953600.06%0.02%99.99%100.00%
30100.01%0.00%100.00%100.00%
Total7,7761,860,480

 

6d6 vs. 6‑Cards Bell Curve

Roll6d6
Freq
6‑Card
Freq
 6d6%6‑Card
%
 6d6
Cum%
6‑Card
Cum%
6100.002%0.000%0.002%0.000%
7600.013%0.000%0.015%0.000%
8217200.045%0.003%0.060%0.003%
9568,6400.120%0.031%0.180%0.034%
1012627,3600.270%0.098%0.450%0.132%
1125278,4800.540%0.281%0.990%0.413%
12456171,3600.977%0.614%1.968%1.027%
13756335,5201.620%1.202%3.588%2.229%
141,161570,9602.488%2.046%6.076%4.275%
151,666903,6003.571%3.238%9.647%7.513%
162,2471,289,5204.816%4.621%14.463%12.134%
172,8561,732,3206.121%6.207%20.585%18.341%
183,4312,148,4807.354%7.699%27.939%26.040%
193,9062,517,1208.372%9.020%36.310%35.059%
204,2212,746,8009.047%9.843%45.358%44.902%
214,3322,845,4409.285%10.196%54.642%55.098%
224,2212,746,8009.047%9.843%63.690%64.941%
233,9062,517,1208.372%9.020%72.061%73.960%
243,4312,148,4807.354%7.699%79.415%81.659%
252,8561,732,3206.121%6.207%85.537%87.866%
262,2471,289,5204.816%4.621%90.353%92.487%
271,666903,6003.571%3.238%93.924%95.725%
281,161570,9602.488%2.046%96.412%97.771%
29756335,5201.620%1.202%98.032%98.973%
30456171,3600.977%0.614%99.010%99.587%
3125278,4800.540%0.281%99.550%99.868%
3212627,3600.270%0.098%99.820%99.966%
335686400.120%0.031%99.940%99.997%
34217200.045%0.003%99.985%100.000%
35600.013%0.000%99.998%100.000%
36100.002%0.000%100.000%100.000%
Total46,65627,907,200

 

7d6 vs. 7‑Cards Bell Curve

Roll7d6
Freq
7‑Card
Freq
 7d6%7‑Card
%
 7d6
Cum%
7‑Card
Cum%
7100.0004%0.0000%0.0004%0.0000%
8700.0025%0.0000%0.0029%0.0000%
92800.0100%0.0000%0.0129%0.0000%
108400.0300%0.0000%0.0429%0.0000%
1121020,1600.0750%0.0052%0.1179%0.0052%
12462110,8800.1650%0.0284%0.2829%0.0335%
13917378,0000.3276%0.0967%0.6105%0.1303%
141,667962,6400.5955%0.2464%1.2060%0.3767%
152,8072,121,8401.0027%0.5431%2.2087%0.9198%
164,4174,037,0401.5779%1.0333%3.7866%1.9530%
176,5386,945,1202.3355%1.7776%6.1221%3.7307%
189,14210,891,4403.2657%2.7877%9.3879%6.5183%
1912,11715,800,4004.3285%4.0441%13.7163%10.5624%
2015,26721,299,0405.4537%5.4515%19.1701%16.0139%
2118,32726,948,8806.5469%6.8976%25.7169%22.9115%
2220,99332,014,0807.4992%8.1940%33.2162%31.1055%
2322,96735,864,6408.2044%9.1796%41.4205%40.2851%
2424,01737,956,2408.5795%9.7149%50.0000%50.0000%
2524,01737,956,2408.5795%9.7149%58.5795%59.7149%
2622,96735,864,6408.2044%9.1796%66.7838%68.8945%
2720,99332,014,0807.4992%8.1940%74.2831%77.0885%
2818,32726,948,8806.5469%6.8976%80.8299%83.9861%
2915,26721,299,0405.4537%5.4515%86.2837%89.4376%
3012,11715,800,4004.3285%4.0441%90.6121%93.4817%
319,14210,891,4403.2657%2.7877%93.8779%96.2693%
326,5386,945,1202.3355%1.7776%96.2134%98.0470%
334,4174,037,0401.5779%1.0333%97.7913%99.0802%
342,8072,121,8401.0027%0.5431%98.7940%99.6233%
351,667962,6400.5955%0.2464%99.3895%99.8697%
36917378,0000.3276%0.0967%99.7171%99.9665%
37462110,8800.1650%0.0284%99.8821%99.9948%
3821020,1600.0750%0.0052%99.9571%100.0000%
398400.0300%0.0000%99.9871%100.0000%
402800.0100%0.0000%99.9971%100.0000%
41700.0025%0.0000%99.9996%100.0000%
42100.0004%0.0000%100.0000%100.0000%
Total279,936390,700,800

 

8d6 vs. 8‑Cards Bell Curve

Roll8d6
Freq
8‑Card
Freq
 8d6%8‑Card
%
 8d6
Cum%
8‑Card
Cum%
8100.0001%0.0000%0.0001%0.0000%
9800.0005%0.0000%0.0005%0.0000%
103600.0021%0.0000%0.0027%0.0000%
1112000.0071%0.0000%0.0098%0.0000%
1233000.0196%0.0000%0.0295%0.0000%
1379200.0472%0.0000%0.0766%0.0000%
141,708241,9200.1017%0.0048%0.1783%0.0048%
153,3681,128,9600.2005%0.0222%0.3788%0.0270%
166,1474,072,3200.3660%0.0802%0.7448%0.1072%
1710,48010,160,6400.6240%0.2000%1.3688%0.3072%
1816,80822,700,1601.0007%0.4469%2.3695%0.7541%
1925,48843,424,6401.5175%0.8550%3.8870%1.6091%
2036,68876,003,2002.1843%1.4964%6.0713%3.1055%
2150,288120,153,6002.9940%2.3656%9.0653%5.4711%
2265,808177,649,9203.9180%3.4977%12.9833%8.9688%
2382,384242,847,3604.9049%4.7813%17.8883%13.7501%
2498,813313,608,9605.8831%6.1745%23.7713%19.9246%
25113,688379,209,6006.7687%7.4661%30.5400%27.3906%
26125,588435,254,4007.4772%8.5695%38.0172%35.9601%
27133,288470,736,0007.9356%9.2681%45.9528%45.2282%
28135,954484,727,0408.0944%9.5435%54.0472%54.7718%
29133,288470,736,0007.9356%9.2681%61.9828%64.0399%
30125,588435,254,4007.4772%8.5695%69.4600%72.6094%
31113,688379,209,6006.7687%7.4661%76.2287%80.0754%
3298,813313,608,9605.8831%6.1745%82.1117%86.2499%
3382,384242,847,3604.9049%4.7813%87.0167%91.0312%
3465,808177,649,9203.9180%3.4977%90.9347%94.5289%
3550,288120,153,6002.9940%2.3656%93.9287%96.8945%
3636,68876,003,2002.1843%1.4964%96.1130%98.3909%
3725,48843,424,6401.5175%0.8550%97.6305%99.2459%
3816,80822,700,1601.0007%0.4469%98.6312%99.6928%
3910,48010,160,6400.6240%0.2000%99.2552%99.8928%
406,1474,072,3200.3660%0.0802%99.6212%99.9730%
413,3681,128,9600.2005%0.0222%99.8217%99.9952%
421,708241,9200.1017%0.0048%99.9234%100.0000%
4379200.0472%0.0000%99.9705%100.0000%
4433000.0196%0.0000%99.9902%100.0000%
4512000.0071%0.0000%99.9973%100.0000%
463600.0021%0.0000%99.9995%100.0000%
47800.0005%0.0000%99.9999%100.0000%
48100.0001%0.0000%100.0000%100.0000%
Total1,679,6165,079,110,400

 

9d6 vs. 9‑Cards Bell Curve

Roll9d6
Freq
9‑Card
Freq
 9d6%9‑Card
%
 9d6
Cum%
9‑Card
Cum%
9100.00001%0.00000%0.00001%0.00000%
10900.00009%0.00000%0.00010%0.00000%
114500.00045%0.00000%0.00055%0.00000%
1216500.00164%0.00000%0.00218%0.00000%
1349500.00491%0.00000%0.00709%0.00000%
141,28700.01277%0.00000%0.01987%0.00000%
152,99400.02971%0.00000%0.04957%0.00000%
166,35400.06305%0.00000%0.11262%0.00000%
1712,4651,451,5200.12369%0.00238%0.23631%0.00238%
1822,8259,797,7600.22649%0.01608%0.46280%0.01846%
1939,30333,384,9600.39000%0.05477%0.85280%0.07323%
2063,99991,808,6400.63506%0.15063%1.48786%0.22386%
2198,979207,930,2400.98216%0.34115%2.47002%0.56502%
22145,899413,320,3201.44774%0.67814%3.91776%1.24315%
23205,560735,920,6402.03975%1.20743%5.95751%2.45058%
24277,4641,201,858,5602.75325%1.97190%8.71076%4.42248%
25359,4691,807,505,2803.56698%2.96559%12.27774%7.38807%
26447,6692,539,434,2404.44218%4.16647%16.71991%11.55454%
27536,5693,342,850,5605.32432%5.48464%22.04424%17.03918%
28619,5694,146,629,7606.14792%6.80341%28.19216%23.84258%
29689,7154,859,688,9606.84398%7.97333%35.03613%31.81591%
30740,6195,398,202,8807.34909%8.85687%42.38522%40.67278%
31767,3945,684,878,0807.61478%9.32722%50.00000%50.00000%
32767,3945,684,878,0807.61478%9.32722%57.61478%59.32722%
33740,6195,398,202,8807.34909%8.85687%64.96387%68.18409%
34689,7154,859,688,9606.84398%7.97333%71.80784%76.15742%
35619,5694,146,629,7606.14792%6.80341%77.95576%82.96082%
36536,5693,342,850,5605.32432%5.48464%83.28009%88.44546%
37447,6692,539,434,2404.44218%4.16647%87.72226%92.61193%
38359,4691,807,505,2803.56698%2.96559%91.28924%95.57752%
39277,4641,201,858,5602.75325%1.97190%94.04249%97.54942%
40205,560735,920,6402.03975%1.20743%96.08224%98.75685%
41145,899413,320,3201.44774%0.67814%97.52998%99.43498%
4298,979207,930,2400.98216%0.34115%98.51214%99.77613%
4363,99991,808,6400.63506%0.15063%99.14720%99.92676%
4439,30333,384,9600.39000%0.05477%99.53720%99.98153%
4522,8259,797,7600.22649%0.01608%99.76369%99.99762%
4612,4651,451,5200.12369%0.00238%99.88738%100.00000%
476,35400.06305%0.00000%99.95043%100.00000%
482,99400.02971%0.00000%99.98013%100.00000%
491,28700.01277%0.00000%99.99290%100.00000%
5049500.00491%0.00000%99.99781%100.00000%
5116500.00164%0.00000%99.99945%100.00000%
524500.00045%0.00000%99.99990%100.00000%
53900.00009%0.00000%99.99999%100.00000%
54100.00001%0.00000%100.00000%100.00000%
Total10,077,696 60,949,324,800

 

10d6 vs. 10‑Cards Bell Curve

11d6 vs. 11‑Cards Bell Curve

12d6 vs. 12‑Cards Bell Curve

Copyright 2018 Eric B. Smith

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Cards as Dice

by Eric B. Smith

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A method that works fairly well is to use a deck of cards instead of dice. The simplest way is to make a deck of 216 cards, each labeled with one of the 216 possible results you can roll on 3d6. You can also write in the 36 possible results on 2d6, doing that six times, and the 6 possible results for 1d6, doing that 36 times. This will allow a 216 card deck correctly simulate a roll of 1d6, 2d6, or 3d6 with perfect accuracy. However, 216 cards is a fairly large deck of cards to handle and shuffle, and to keep it perfectly accurate you'd need to reshuffle it after every die roll. It is certainly too large of a deck to be convieniently used in a traveling situation, such as on a bus or car ride.
20-Card Deck
Qty. Card
1  0 (Joker)
2  1's (Ace)
3 2's
4 3's
4 4's
3 5's
2 6's
1 7

Another method I hit on is to build a smaller deck of just 20‑cards (thanks to a discussion on GURPS Forums, and in particular thanks to Brandy). You can build that 20‑card deck by taking cards numbered 0 to 7, in quantities from 1 to 4 (see table to the right); this small deck can be built using a deck of standard playing cards, using a Joker or face card for the zero and two Aces for the ones. When you make a die roll you simply draw a number of cards equal to the number of dice that need to be rolled and add them together. The resulting bell curve from this 20‑card deck is amazingly close to the bell curve of rolling 3d6, and is very close for 2d6 and 4d6. It is close enough for rolling 1d6 or for rolling more dice, up to 6d6, and will yield fair result with the same average and a similar shaped bell curve – tending more heavily towards the average the more cards you draw – even for rolls such as 9d6 or 12d6.

For 1d6 you could keep a separate pile of cards numbered 1-6 to draw from, but this precludes using one standard playing card deck. Another option is to use the 20 card deck, but place 1-6 all in spades, with the 0 and 7, and all other cards, as another suit. To roll 1d6 you simply draw from the deck until you draw a spade.

At any rate, the draw deck will give a few odd results which you will need to decide how to deal with. You can draw a result of 2 and 19 on 3d6. For skill rolls these should just be treated as a 3 and 18, respectively. For damage rolls you will need to decide whether to threat them as the actual result drawn or to normalize them to 3 and 18. Personally I recommend just leaving them as the result drawn. Similarly, on 1d6 you can draw a 0 or 7, and on 2d6 you can draw a 1 or 13. Again, since these will always be damage rolls I recommend just leaving them as the result that was drawn.

 

Copyright © 2018 Eric B. Smith

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