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
	View 1d6 Data Table +   Roll 1d6 Freq 1‑Card Freq   1d6 % 1‑Card %   1d6 Cum% 1‑Card Cum% 0 0 1 0.00% 5.00% 0.00% 5.00% 1 1 2 16.67% 10.00% 16.67% 15.00% 2 1 3 16.67% 15.00% 33.33% 30.00% 3 1 4 16.67% 20.00% 50.00% 50.00% 4 1 4 16.67% 20.00% 66.67% 70.00% 5 1 3 16.67% 15.00% 83.33% 85.00% 6 1 2 16.67% 10.00% 100.00% 95.00% 7 0 1 0.00% 5.00% 100.00% 100.00% Total 6 20
	2d6 vs. 2‑Cards Bell Curve
	View 2d6 Data Table +   Roll 2d6 Freq 2‑Card Freq   2d6 % 2‑Card %   2d6 Cum% 2‑Card Cum% 1 0 4 0.00% 1.05% 0.00% 1.05% 2 1 8 2.78% 2.11% 2.78% 3.16% 3 2 20 5.56% 5.26% 8.33% 8.42% 4 3 30 8.33% 7.89% 16.67% 16.32% 5 4 46 11.11% 12.11% 27.78% 28.42% 6 5 52 13.89% 13.68% 41.67% 42.11% 7 6 60 16.67% 15.79% 58.33% 57.89% 8 5 52 13.89% 13.68% 72.22% 71.58% 9 4 46 11.11% 12.11% 83.33% 83.68% 10 3 30 8.33% 7.89% 91.67% 91.58% 11 2 20 5.56% 5.26% 97.22% 96.84% 12 1 8 2.78% 2.11% 100.00% 98.95% 13 0 4 0.00% 1.05% 100.00% 100.00% Total 36 380
	3d6 vs. 3‑Cards Bell Curve
	View 3d6 Data Table +   Roll 3d6 Freq 3‑Card Freq   3d6 % 3‑Card %   3d6 Cum% 3‑Card Cum% 2 0 6 0.00% 0.09% 0.00% 0.09% 3 1 36 0.46% 0.53% 0.46% 0.61% 4 3 84 1.39% 1.23% 1.85% 1.84% 5 6 180 2.78% 2.63% 4.63% 4.47% 6 10 318 4.63% 4.65% 9.26% 9.12% 7 15 480 6.94% 7.02% 16.20% 16.14% 8 21 648 9.72% 9.47% 25.93% 25.61% 9 25 798 11.57% 11.67% 37.50% 37.28% 10 27 870 12.50% 12.72% 50.00% 50.00% 11 27 870 12.50% 12.72% 62.50% 62.72% 12 25 798 11.57% 11.67% 74.07% 74.39% 13 21 648 9.72% 9.47% 83.80% 83.86% 14 15 480 6.94% 7.02% 90.74% 90.88% 15 10 318 4.63% 4.65% 95.37% 95.53% 16 6 180 2.78% 2.63% 98.15% 98.16% 17 3 84 1.39% 1.23% 99.54% 99.39% 18 1 36 0.46% 0.53% 100.00% 99.91% 19 0 6 0.00% 0.09% 100.00% 100.00% Total 216 6,840
	4d6 vs. 4‑Cards Bell Curve
	View 4d6 Data Table +   Roll 4d6 Freq 4‑Card Freq   4d6 % 4‑Card %   4d6 Cum% 4‑Card Cum% 4 1 72 0.08% 0.06% 0.08% 0.06% 5 4 240 0.31% 0.21% 0.39% 0.27% 6 10 768 0.77% 0.66% 1.16% 0.93% 7 20 1,560 1.54% 1.34% 2.70% 2.27% 8 35 2,976 2.70% 2.56% 5.40% 4.83% 9 56 4,776 4.32% 4.11% 9.72% 8.94% 10 80 7,152 6.17% 6.15% 15.90% 15.09% 11 104 9,336 8.02% 8.03% 23.92% 23.12% 12 125 11,592 9.65% 9.97% 33.56% 33.09% 13 140 12,888 10.80% 11.08% 44.37% 44.17% 14 146 13,560 11.27% 11.66% 55.63% 55.83% 15 140 12,888 10.80% 11.08% 66.44% 66.91% 16 125 11,592 9.65% 9.97% 76.08% 76.88% 17 104 9,336 8.02% 8.03% 84.10% 84.91% 18 80 7,152 6.17% 6.15% 90.28% 91.06% 19 56 4,776 4.32% 4.11% 94.60% 95.17% 20 35 2,976 2.70% 2.56% 97.30% 97.73% 21 20 1,560 1.54% 1.34% 98.84% 99.07% 22 10 768 0.77% 0.66% 99.61% 99.73% 23 4 240 0.31% 0.21% 99.92% 99.94% 24 1 72 0.08% 0.06% 100.00% 100.00% Total 1,296 116,280
	5d6 vs. 5‑Cards Bell Curve
	View 5d6 Data Table +   Roll 5d6 Freq 5‑Card Freq   5d6 % 5‑Card %   5d6 Cum% 5‑Card Cum% 5 1 0 0.01% 0.00% 0.01% 0.00% 6 5 360 0.06% 0.02% 0.08% 0.02% 7 15 1,680 0.19% 0.09% 0.27% 0.11% 8 35 5,160 0.45% 0.28% 0.72% 0.39% 9 70 12,120 0.90% 0.65% 1.62% 1.04% 10 126 24,720 1.62% 1.33% 3.24% 2.37% 11 205 43,080 2.64% 2.32% 5.88% 4.68% 12 305 68,400 3.92% 3.68% 9.80% 8.36% 13 420 98,640 5.40% 5.30% 15.20% 13.66% 14 540 131,160 6.94% 7.05% 22.15% 20.71% 15 651 161,520 8.37% 8.68% 30.52% 29.39% 16 735 185,280 9.45% 9.96% 39.97% 39.35% 17 780 198,120 10.03% 10.65% 50.00% 50.00% 18 780 198,120 10.03% 10.65% 60.03% 60.65% 19 735 185,280 9.45% 9.96% 69.48% 70.61% 20 651 161,520 8.37% 8.68% 77.85% 79.29% 21 540 131,160 6.94% 7.05% 84.80% 86.34% 22 420 98,640 5.40% 5.30% 90.20% 91.64% 23 305 68,400 3.92% 3.68% 94.12% 95.32% 24 205 43,080 2.64% 2.32% 96.76% 97.63% 25 126 24,720 1.62% 1.33% 98.38% 98.96% 26 70 12,120 0.90% 0.65% 99.28% 99.61% 27 35 5,160 0.45% 0.28% 99.73% 99.89% 28 15 1,680 0.19% 0.09% 99.92% 99.98% 29 5 360 0.06% 0.02% 99.99% 100.00% 30 1 0 0.01% 0.00% 100.00% 100.00% Total 7,776 1,860,480
	6d6 vs. 6‑Cards Bell Curve
	View 6d6 Data Table +   Roll 6d6 Freq 6‑Card Freq   6d6% 6‑Card %   6d6 Cum% 6‑Card Cum% 6 1 0 0.002% 0.000% 0.002% 0.000% 7 6 0 0.013% 0.000% 0.015% 0.000% 8 21 720 0.045% 0.003% 0.060% 0.003% 9 56 8,640 0.120% 0.031% 0.180% 0.034% 10 126 27,360 0.270% 0.098% 0.450% 0.132% 11 252 78,480 0.540% 0.281% 0.990% 0.413% 12 456 171,360 0.977% 0.614% 1.968% 1.027% 13 756 335,520 1.620% 1.202% 3.588% 2.229% 14 1,161 570,960 2.488% 2.046% 6.076% 4.275% 15 1,666 903,600 3.571% 3.238% 9.647% 7.513% 16 2,247 1,289,520 4.816% 4.621% 14.463% 12.134% 17 2,856 1,732,320 6.121% 6.207% 20.585% 18.341% 18 3,431 2,148,480 7.354% 7.699% 27.939% 26.040% 19 3,906 2,517,120 8.372% 9.020% 36.310% 35.059% 20 4,221 2,746,800 9.047% 9.843% 45.358% 44.902% 21 4,332 2,845,440 9.285% 10.196% 54.642% 55.098% 22 4,221 2,746,800 9.047% 9.843% 63.690% 64.941% 23 3,906 2,517,120 8.372% 9.020% 72.061% 73.960% 24 3,431 2,148,480 7.354% 7.699% 79.415% 81.659% 25 2,856 1,732,320 6.121% 6.207% 85.537% 87.866% 26 2,247 1,289,520 4.816% 4.621% 90.353% 92.487% 27 1,666 903,600 3.571% 3.238% 93.924% 95.725% 28 1,161 570,960 2.488% 2.046% 96.412% 97.771% 29 756 335,520 1.620% 1.202% 98.032% 98.973% 30 456 171,360 0.977% 0.614% 99.010% 99.587% 31 252 78,480 0.540% 0.281% 99.550% 99.868% 32 126 27,360 0.270% 0.098% 99.820% 99.966% 33 56 8640 0.120% 0.031% 99.940% 99.997% 34 21 720 0.045% 0.003% 99.985% 100.000% 35 6 0 0.013% 0.000% 99.998% 100.000% 36 1 0 0.002% 0.000% 100.000% 100.000% Total 46,656 27,907,200
	7d6 vs. 7‑Cards Bell Curve
	View 7d6 Data Table +   Roll 7d6 Freq 7‑Card Freq   7d6% 7‑Card %   7d6 Cum% 7‑Card Cum% 7 1 0 0.0004% 0.0000% 0.0004% 0.0000% 8 7 0 0.0025% 0.0000% 0.0029% 0.0000% 9 28 0 0.0100% 0.0000% 0.0129% 0.0000% 10 84 0 0.0300% 0.0000% 0.0429% 0.0000% 11 210 20,160 0.0750% 0.0052% 0.1179% 0.0052% 12 462 110,880 0.1650% 0.0284% 0.2829% 0.0335% 13 917 378,000 0.3276% 0.0967% 0.6105% 0.1303% 14 1,667 962,640 0.5955% 0.2464% 1.2060% 0.3767% 15 2,807 2,121,840 1.0027% 0.5431% 2.2087% 0.9198% 16 4,417 4,037,040 1.5779% 1.0333% 3.7866% 1.9530% 17 6,538 6,945,120 2.3355% 1.7776% 6.1221% 3.7307% 18 9,142 10,891,440 3.2657% 2.7877% 9.3879% 6.5183% 19 12,117 15,800,400 4.3285% 4.0441% 13.7163% 10.5624% 20 15,267 21,299,040 5.4537% 5.4515% 19.1701% 16.0139% 21 18,327 26,948,880 6.5469% 6.8976% 25.7169% 22.9115% 22 20,993 32,014,080 7.4992% 8.1940% 33.2162% 31.1055% 23 22,967 35,864,640 8.2044% 9.1796% 41.4205% 40.2851% 24 24,017 37,956,240 8.5795% 9.7149% 50.0000% 50.0000% 25 24,017 37,956,240 8.5795% 9.7149% 58.5795% 59.7149% 26 22,967 35,864,640 8.2044% 9.1796% 66.7838% 68.8945% 27 20,993 32,014,080 7.4992% 8.1940% 74.2831% 77.0885% 28 18,327 26,948,880 6.5469% 6.8976% 80.8299% 83.9861% 29 15,267 21,299,040 5.4537% 5.4515% 86.2837% 89.4376% 30 12,117 15,800,400 4.3285% 4.0441% 90.6121% 93.4817% 31 9,142 10,891,440 3.2657% 2.7877% 93.8779% 96.2693% 32 6,538 6,945,120 2.3355% 1.7776% 96.2134% 98.0470% 33 4,417 4,037,040 1.5779% 1.0333% 97.7913% 99.0802% 34 2,807 2,121,840 1.0027% 0.5431% 98.7940% 99.6233% 35 1,667 962,640 0.5955% 0.2464% 99.3895% 99.8697% 36 917 378,000 0.3276% 0.0967% 99.7171% 99.9665% 37 462 110,880 0.1650% 0.0284% 99.8821% 99.9948% 38 210 20,160 0.0750% 0.0052% 99.9571% 100.0000% 39 84 0 0.0300% 0.0000% 99.9871% 100.0000% 40 28 0 0.0100% 0.0000% 99.9971% 100.0000% 41 7 0 0.0025% 0.0000% 99.9996% 100.0000% 42 1 0 0.0004% 0.0000% 100.0000% 100.0000% Total 279,936 390,700,800
	8d6 vs. 8‑Cards Bell Curve
	View 8d6 Data Table +   Roll 8d6 Freq 8‑Card Freq   8d6% 8‑Card %   8d6 Cum% 8‑Card Cum% 8 1 0 0.0001% 0.0000% 0.0001% 0.0000% 9 8 0 0.0005% 0.0000% 0.0005% 0.0000% 10 36 0 0.0021% 0.0000% 0.0027% 0.0000% 11 120 0 0.0071% 0.0000% 0.0098% 0.0000% 12 330 0 0.0196% 0.0000% 0.0295% 0.0000% 13 792 0 0.0472% 0.0000% 0.0766% 0.0000% 14 1,708 241,920 0.1017% 0.0048% 0.1783% 0.0048% 15 3,368 1,128,960 0.2005% 0.0222% 0.3788% 0.0270% 16 6,147 4,072,320 0.3660% 0.0802% 0.7448% 0.1072% 17 10,480 10,160,640 0.6240% 0.2000% 1.3688% 0.3072% 18 16,808 22,700,160 1.0007% 0.4469% 2.3695% 0.7541% 19 25,488 43,424,640 1.5175% 0.8550% 3.8870% 1.6091% 20 36,688 76,003,200 2.1843% 1.4964% 6.0713% 3.1055% 21 50,288 120,153,600 2.9940% 2.3656% 9.0653% 5.4711% 22 65,808 177,649,920 3.9180% 3.4977% 12.9833% 8.9688% 23 82,384 242,847,360 4.9049% 4.7813% 17.8883% 13.7501% 24 98,813 313,608,960 5.8831% 6.1745% 23.7713% 19.9246% 25 113,688 379,209,600 6.7687% 7.4661% 30.5400% 27.3906% 26 125,588 435,254,400 7.4772% 8.5695% 38.0172% 35.9601% 27 133,288 470,736,000 7.9356% 9.2681% 45.9528% 45.2282% 28 135,954 484,727,040 8.0944% 9.5435% 54.0472% 54.7718% 29 133,288 470,736,000 7.9356% 9.2681% 61.9828% 64.0399% 30 125,588 435,254,400 7.4772% 8.5695% 69.4600% 72.6094% 31 113,688 379,209,600 6.7687% 7.4661% 76.2287% 80.0754% 32 98,813 313,608,960 5.8831% 6.1745% 82.1117% 86.2499% 33 82,384 242,847,360 4.9049% 4.7813% 87.0167% 91.0312% 34 65,808 177,649,920 3.9180% 3.4977% 90.9347% 94.5289% 35 50,288 120,153,600 2.9940% 2.3656% 93.9287% 96.8945% 36 36,688 76,003,200 2.1843% 1.4964% 96.1130% 98.3909% 37 25,488 43,424,640 1.5175% 0.8550% 97.6305% 99.2459% 38 16,808 22,700,160 1.0007% 0.4469% 98.6312% 99.6928% 39 10,480 10,160,640 0.6240% 0.2000% 99.2552% 99.8928% 40 6,147 4,072,320 0.3660% 0.0802% 99.6212% 99.9730% 41 3,368 1,128,960 0.2005% 0.0222% 99.8217% 99.9952% 42 1,708 241,920 0.1017% 0.0048% 99.9234% 100.0000% 43 792 0 0.0472% 0.0000% 99.9705% 100.0000% 44 330 0 0.0196% 0.0000% 99.9902% 100.0000% 45 120 0 0.0071% 0.0000% 99.9973% 100.0000% 46 36 0 0.0021% 0.0000% 99.9995% 100.0000% 47 8 0 0.0005% 0.0000% 99.9999% 100.0000% 48 1 0 0.0001% 0.0000% 100.0000% 100.0000% Total 1,679,616 5,079,110,400
	9d6 vs. 9‑Cards Bell Curve
	View 9d6 Data Table +   Roll 9d6 Freq 9‑Card Freq   9d6% 9‑Card %   9d6 Cum% 9‑Card Cum% 9 1 0 0.00001% 0.00000% 0.00001% 0.00000% 10 9 0 0.00009% 0.00000% 0.00010% 0.00000% 11 45 0 0.00045% 0.00000% 0.00055% 0.00000% 12 165 0 0.00164% 0.00000% 0.00218% 0.00000% 13 495 0 0.00491% 0.00000% 0.00709% 0.00000% 14 1,287 0 0.01277% 0.00000% 0.01987% 0.00000% 15 2,994 0 0.02971% 0.00000% 0.04957% 0.00000% 16 6,354 0 0.06305% 0.00000% 0.11262% 0.00000% 17 12,465 1,451,520 0.12369% 0.00238% 0.23631% 0.00238% 18 22,825 9,797,760 0.22649% 0.01608% 0.46280% 0.01846% 19 39,303 33,384,960 0.39000% 0.05477% 0.85280% 0.07323% 20 63,999 91,808,640 0.63506% 0.15063% 1.48786% 0.22386% 21 98,979 207,930,240 0.98216% 0.34115% 2.47002% 0.56502% 22 145,899 413,320,320 1.44774% 0.67814% 3.91776% 1.24315% 23 205,560 735,920,640 2.03975% 1.20743% 5.95751% 2.45058% 24 277,464 1,201,858,560 2.75325% 1.97190% 8.71076% 4.42248% 25 359,469 1,807,505,280 3.56698% 2.96559% 12.27774% 7.38807% 26 447,669 2,539,434,240 4.44218% 4.16647% 16.71991% 11.55454% 27 536,569 3,342,850,560 5.32432% 5.48464% 22.04424% 17.03918% 28 619,569 4,146,629,760 6.14792% 6.80341% 28.19216% 23.84258% 29 689,715 4,859,688,960 6.84398% 7.97333% 35.03613% 31.81591% 30 740,619 5,398,202,880 7.34909% 8.85687% 42.38522% 40.67278% 31 767,394 5,684,878,080 7.61478% 9.32722% 50.00000% 50.00000% 32 767,394 5,684,878,080 7.61478% 9.32722% 57.61478% 59.32722% 33 740,619 5,398,202,880 7.34909% 8.85687% 64.96387% 68.18409% 34 689,715 4,859,688,960 6.84398% 7.97333% 71.80784% 76.15742% 35 619,569 4,146,629,760 6.14792% 6.80341% 77.95576% 82.96082% 36 536,569 3,342,850,560 5.32432% 5.48464% 83.28009% 88.44546% 37 447,669 2,539,434,240 4.44218% 4.16647% 87.72226% 92.61193% 38 359,469 1,807,505,280 3.56698% 2.96559% 91.28924% 95.57752% 39 277,464 1,201,858,560 2.75325% 1.97190% 94.04249% 97.54942% 40 205,560 735,920,640 2.03975% 1.20743% 96.08224% 98.75685% 41 145,899 413,320,320 1.44774% 0.67814% 97.52998% 99.43498% 42 98,979 207,930,240 0.98216% 0.34115% 98.51214% 99.77613% 43 63,999 91,808,640 0.63506% 0.15063% 99.14720% 99.92676% 44 39,303 33,384,960 0.39000% 0.05477% 99.53720% 99.98153% 45 22,825 9,797,760 0.22649% 0.01608% 99.76369% 99.99762% 46 12,465 1,451,520 0.12369% 0.00238% 99.88738% 100.00000% 47 6,354 0 0.06305% 0.00000% 99.95043% 100.00000% 48 2,994 0 0.02971% 0.00000% 99.98013% 100.00000% 49 1,287 0 0.01277% 0.00000% 99.99290% 100.00000% 50 495 0 0.00491% 0.00000% 99.99781% 100.00000% 51 165 0 0.00164% 0.00000% 99.99945% 100.00000% 52 45 0 0.00045% 0.00000% 99.99990% 100.00000% 53 9 0 0.00009% 0.00000% 99.99999% 100.00000% 54 1 0 0.00001% 0.00000% 100.00000% 100.00000% Total 10,077,696 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 Back to Top Back to GURPSLand

Cards as Dice

by Eric B. Smith

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A fun experiment in gaming is to replace the dice rolls with other methods. I suggest GURPS with Polyhedral dice in another article, but a good alternative to use while traveling is to use a deck of cards.

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 to 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 the 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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