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Why people think bots cheat

Posted By: Jason Lee
Date: Tuesday, 24 August 2010, at 4:12 p.m.

In Response To: Why people think bots cheat (Timothy Chow)

Using Wolfram Alpha:

((x^24 + x^20 + x^16 + x^12 + 2*x^11 + 2x^10 + 4*x^9 + 5*x^8 + 6*x^7 + 4*x^6 + 4*x^5 + 3*x^4 + 2*x^3)^9 =

x^216+9 x^212+45 x^208+165 x^204+18 x^203+18 x^202+36 x^201+531 x^200+198 x^199+180 x^198+324 x^197+1593 x^196+1098 x^195+936 x^194+1584 x^193+4434 x^192+4248 x^191+3600 x^190+5904 x^189+12042 x^188+14220 x^187+13284 x^186+20808 x^185+35001 x^184+46044 x^183+48456 x^182+70344 x^181+105085 x^180+140508 x^179+159048 x^178+214800 x^177+299223 x^176+393192 x^175+465648 x^174+604080 x^173+815751 x^172+1062462 x^171+1309446 x^170+1679148 x^169+2232927 x^168+2891538 x^167+3634524 x^166+4613772 x^165+5997393 x^164+7644006 x^163+9565152 x^162+11955744 x^161+15162408 x^160+19020816 x^159+23614848 x^158+29293632 x^157+36706776 x^156+45842256 x^155+56930400 x^154+70694016 x^153+88213581 x^152+109867104 x^151+136048704 x^150+168055200 x^149+207484677 x^148+255376032 x^147+312441984 x^146+380988000 x^145+463753513 x^144+563387904 x^143+682141248 x^142+824806656 x^141+997208433 x^140+1205565066 x^139+1455806826 x^138+1756848564 x^137+2119236759 x^136+2553551118 x^135+3069967140 x^134+3681233748 x^133+4402664037 x^132+5249465154 x^131+6237789192 x^130+7387710768 x^129+8725742082 x^128+10281519288 x^127+12090254640 x^126+14195288880 x^125+16651397610 x^124+19519543068 x^123+22867257204 x^122+26768952072 x^121+31306738275 x^120+36563399532 x^119+42620095080 x^118+49558132008 x^117+57461243895 x^116+66414691548 x^115+76507494744 x^114+87843500496 x^113+100550268369 x^112+114785856504 x^111+130741852464 x^110+148652430192 x^109+168793608193 x^108+191475134658 x^107+217022047578 x^106+245759409684 x^105+277989695289 x^104+313967697966 x^103+353877064548 x^102+397822072788 x^101+445831639767 x^100+497873959178 x^99+553886206416 x^98+613819530144 x^97+677692374840 x^96+745634129184 x^95+817916450976 x^94+894961190688 x^93+977319569160 x^92+1065608150112 x^91+1160411539616 x^90+1262159975520 x^89+1371000357531 x^88+1486673524032 x^87+1608425627328 x^86+1734976503936 x^85+1864560587043 x^84+1995041482128 x^83+2124093431952 x^82+2249432716864 x^81+2369064097935 x^80+2481501044880 x^79+2585916396864 x^78+2682192944448 x^77+2770849037511 x^76+2852836494246 x^75+2929227880950 x^74+3000837965868 x^73+3067835222837 x^72+3129409816722 x^71+3183561651420 x^70+3227062331436 x^69+3255618814839 x^68+3264235760574 x^67+3247741856856 x^66+3201417174960 x^65+3121637908194 x^64+3006445496040 x^63+2855955944880 x^62+2672545957488 x^61+2460785006778 x^60+2227117856220 x^59+1979339566500 x^58+1725933378984 x^57+1475359262559 x^56+1235382229692 x^55+1012518788680 x^54+811657156392 x^53+635878015155 x^52+486472174092 x^51+363125919384 x^50+264227055984 x^49+187236340053 x^48+129070228536 x^47+86449863024 x^46+56185231600 x^45+35378722485 x^44+21546475146 x^43+12666779010 x^42+7171637508 x^41+3900051981 x^40+2030715366 x^39+1008582516 x^38+475624260 x^37+211760067 x^36+88379730 x^35+34260912 x^34+12186912 x^33+3911328 x^32+1105344 x^31+264960 x^30+50688 x^29+6912 x^28+512 x^27

(yikes!!!)

I ignored the /36 you put in (for now). So, summing the relevant coefficients:

21546475146 + 12666779010 + 7171637508 + 3900051981 + 2030715366 + 1008582516 + 475624260 + 211760067 + 88379730 + 34260912 + 12186912 + 3911328 + 1105344 + 264960 + 50688 + 6912 + 512 = 49151793152

Note that each of these coefficients has an interpretation. For example, the 50688 coefficient on x^29 represents the number of ways of rolling EXACTLY 29 pips in nine consecutive rolls of differently colored dice, so that 12 is different from 12. I guess I should also mention that the coefficient on the original polynomial has an interpretation too -- for example, the 5 on 5x^8 represents the number of ways of rolling exactly 8 pips. Well, we all knew that already, it can be done with 26, 35, 22 -- five ways.

Now we divide this by 36^9 = 101559956668416 to get

49151793152 / 101559956668416

Which reduces to

23999899 / 49589822592

which is approximately 0.0004839682367379903854284794937626543546074559830520476164...

Take the reciprocal to get that this happens about 1 out of every 2066 nine-roll sequences. So, a bit more rare than rolling consecutive boxes.

Please don't be impressed at any of this... Timothy Chow was the one who posted the generating function solution, I just did the grubby work to get numbers.

JLee

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