When Not To Bear Off

From: Ian Shaw   

Interesting article, Walter, and congratulations on your win in Vegas. Is it possible to publish The List as a table rather than as a gif file, please? At present, I can't get it to print out more than the 1st pageful. As a table, it would even be possible to paste it into excel for fiddling with.

From: Michael Strato   

Hi Ian and Walter,

I am back from the Caribbean. Walter, congratulations on winning Vegas! Wow! And thanks for taking care of Ian. If anyone else want the text version, please copy and paste it from below.

Best regards,

Michael Strato GV Editor

Smoothing Deuces 517100 4/2 0.544053 418100 4/2 0.514937 616100 4/2 0.482603 319100 4/2 0.465515 715100 4/2 0.398654 317100 4/2 0.310501 516200 4/2 0.296612 515100 4/2 0.286081 416100 4/2 0.279129 417200 4/2 0.266332 318200 4/2 0.242942 615200 4/2 0.227450 418002 6/4 0.198694 517002 6/4 0.194578 814100 4/2 0.182317 616002 6/4 0.162243 21a100 4/2 0.156302 614100 4/2 0.155292 319002 6/4 0.150481 714200 4/2 0.133297 514031 6/4 0.122431 415031 6/4 0.106780 316200 4/2 0.106511 315100 4/2 0.105702 715001 6/4 0.102794 516002 6/4 0.101906 417002 6/4 0.098365 510710 5/3 0.091792 314051 6/4 0.086375 316031 6/4 0.085013 613031 6/4 0.083210 515300 4/2 0.082586 318002 6/4 0.082309 313061 6/4 0.081831 414041 6/4 0.081516 218100 4/2 0.080345 715002 6/4 0.078709 416300 4/2 0.077985 315041 6/4 0.073946 413051 6/4 0.068860 416002 6/4 0.067829 410810 5/3 0.062148 610610 5/3 0.061509 317002 6/4 0.061457 415200 4/2 0.058110 513041 6/4 0.057554 616001 6/4 0.056251 314061 6/4 0.055684 317300 4/2 0.055303 710510 5/3 0.054950 313071 6/4 0.052526 214071 6/4 0.051132 515021 6/4 0.049513 214061 6/4 0.048606 733100 4/2 0.043879 514200 4/2 0.043484 517001 6/4 0.043073 412061 6/4 0.042364 215061 6/4 0.038195 213071 6/4 0.037144 310910 5/3 0.036907 314031 6/4 0.035360 615002 6/4 0.035126 215051 6/4 0.035071 21a002 6/4 0.034749 913100 4/2 0.034451 515002 6/4 0.034248 515001 6/4 0.033489 312071 6/4 0.030169 213081 6/4 0.030120 413061 6/4 0.025233 713100 4/2 0.025171 512051 6/4 0.020236 416021 6/4 0.017504 510620 5/3 0.017221 413031 6/4 0.015846 315051 6/4 0.015480 614021 6/4 0.015172 312081 6/4 0.014528 510510 5/3 0.014279 412071 6/4 0.006549 414051 6/4 0.005148 316002 6/4 0.004726 316013 6/4 0.004244 832100 4/2 0.003807 a12100 4/2 0.002545 219200 4/2 0.002251 625100 4/2 0.000845

Smoothing Threes 431501 6/3 0.540175 421601 6/3 0.539788 331601 6/3 0.530788 521501 6/3 0.518218 401901 6/3 0.516780 401802 6/3 0.511234 321701 6/3 0.503291 301902 6/3 0.480415 501801 6/3 0.472499 531401 6/3 0.465578 301a01 6/3 0.457331 341501 6/3 0.453333 501702 6/3 0.445449 621401 6/3 0.416888 311802 6/3 0.404117 441401 6/3 0.401723 411702 6/3 0.399648 601701 6/3 0.355446 601602 6/3 0.323280 201a02 6/3 0.321128 511602 6/3 0.320947 231701 6/3 0.320187 631301 6/3 0.318541 301802 6/3 0.317554 351401 6/3 0.312493 541301 6/3 0.310799 311801 6/3 0.308085 411701 6/3 0.305407 221801 6/3 0.298417 321501 6/3 0.298159 311901 6/3 0.292708 321702 6/3 0.285373 401702 6/3 0.282529 211902 6/3 0.279846 411801 6/3 0.267507 511601 6/3 0.258896 241601 6/3 0.251709 201b01 6/3 0.251366 421602 6/3 0.244288 421401 6/3 0.242931 721301 6/3 0.238098 451301 6/3 0.235283 331401 6/3 0.234576 221802 6/3 0.230577 311702 6/3 0.229538 301702 6/3 0.229086 301801 6/3 0.226752 201902 6/3 0.218467 401701 6/3 0.195048 401602 6/3 0.187232 231702 6/3 0.182499 501602 6/3 0.181384 411602 6/3 0.177615 201802 6/3 0.175039 331602 6/3 0.174626 211a01 6/3 0.169768 211901 6/3 0.169153 611502 6/3 0.168721 221601 6/3 0.168611 211802 6/3 0.168383 641201 6/3 0.166001 321602 6/3 0.160015 301901 6/3 0.153857 701502 6/3 0.148464 521301 6/3 0.146300 611501 6/3 0.142840 251501 6/3 0.141684 551201 6/3 0.133426 431301 6/3 0.132847 701601 6/3 0.132165 511701 6/3 0.131396 731201 6/3 0.131175 221702 6/3 0.128670 521502 6/3 0.127952 231501 6/3 0.127291 301803 6/3 0.122388 201901 6/3 0.115614 311602 6/3 0.114734 321801 6/3 0.110624 361301 6/3 0.099923 401703 6/3 0.096197 241602 6/3 0.094308 211702 6/3 0.092844 341301 6/3 0.092340 501710 5/2 0.091792 431502 6/3 0.086883 501502 6/3 0.084691 401801 6/3 0.084452 201903 6/3 0.084360 221901 6/3 0.083164 421502 6/3 0.079970 311601 6/3 0.078911 201a01 6/3 0.066534 821201 6/3 0.064489 401810 5/2 0.062148 601610 5/2 0.061509 231602 6/3 0.060076 701510 5/2 0.054950 331502 6/3 0.048172 511502 6/3 0.047971 501601 6/3 0.047056 311703 6/3 0.046705 341502 6/3 0.044529 411502 6/3 0.038640 621201 6/3 0.038538 301910 5/2 0.036907 211803 6/3 0.031407 531201 6/3 0.030562 211701 6/3 0.018792 231801 6/3 0.018327 301704 6/3 0.018127 501620 5/2 0.017221 201804 6/3 0.015960 501510 5/2 0.014279 301703 6/3 0.011087 301602 6/3 0.009869 411501 6/3 0.008438 201702 6/3 0.007476 711401 6/3 0.007315 461201 6/3 0.002836 651101 6/3 0.001429

Smoothing Fours 408102 6/2 0.198694 507102 6/2 0.194578 606102 6/2 0.162243 309102 6/2 0.150481 504131 6/2 0.122431 405131 6/2 0.106780 705101 6/2 0.102794 506102 6/2 0.101906 407102 6/2 0.098365 304151 6/2 0.086375 306131 6/2 0.085013 603131 6/2 0.083210 308102 6/2 0.082309 303161 6/2 0.081831 404141 6/2 0.081516 705102 6/2 0.078709 305141 6/2 0.073946 403151 6/2 0.068860 406102 6/2 0.067829 307102 6/2 0.061457 503141 6/2 0.057554 606101 6/2 0.056251 304161 6/2 0.055684 303171 6/2 0.052526 204171 6/2 0.051132 505121 6/2 0.049513 204161 6/2 0.048606 507101 6/2 0.043073 402161 6/2 0.042364 205161 6/2 0.038195 203171 6/2 0.037144 304131 6/2 0.035360 605102 6/2 0.035126 205151 6/2 0.035071 20a102 6/2 0.034749 505102 6/2 0.034248 505101 6/2 0.033489 302171 6/2 0.030169 203181 6/2 0.030120 403161 6/2 0.025233 502151 6/2 0.020236 406121 6/2 0.017504 403131 6/2 0.015846 305151 6/2 0.015480 604121 6/2 0.015172 302181 6/2 0.014528 402171 6/2 0.006549 404151 6/2 0.005148 306102 6/2 0.004726 306113 6/2 0.004244

Ray, I do believe your questions deserve an entire article, especially if I am going to try to characterise the Type B plays :-)

As I recall, when I printed out the Type B's there were 20 pages, vs. about 3 pages for the Type A's. As for Type C's, I know of none but I've never really searched. If you look at The List you'll spot a few instances where the correct play of a deuces smooths, producing another position where the correct deuce smooths. A type C, I believe, would have to consist of a linked pair of positions in The List with different numbers to play. I haven't noticed any.

Just completing my analysis of the non-existence of Type C positions ...

Suppose we had two remaining fours to play and had arrived (not, I think, by optimal play) at one of the positions on The List. It would not be correct to play both fours by smoothing. All correct smoothing fours are played 6/2 and a necessary condition for a smoothing four is that there is no checker on the deuce point. After playing one smoothing four there would be one checker on the deuce point. Hence the necessary condition would not be met for playing a second smoothing four.

Suppose we had two remaining threes to play and had arrived (not, I think, by optimal play) at one of the positions on The List. It would not be correct to play both threes by smoothing. All correct smoothing threes require exactly one checker on the three point. But to have the option of playing two smoothing threes instead of bearing off two threes, there would by definition be at least two checkers on the three point. Hence the existence of the option would simultaneously indicate that to play the first three by smoothing (either 6/3 or 5/2) would be wrong.

Finally suppose we had two remaining deuces to play and had arrived (not, I think, by optimal play) at one of the positions on The List. There are only three positions with at least two checkers on the deuce point, a necessary condition to have the option of playing two smoothing deuces instead of bearing off two deuces. In none of these three positions does the first correct smoothing play result in a position where a second smoothing would be correct.

PS: It is amusing to note that the famous Magriel position (515100) is itself one of only five positions which can result from a prior correct smoothing (515001).

From: robadams   

Actually the game position would have been "205102" so quite a ways from "20a102" on the list. But "junior" still says that smoothing was correct.