Almost anyone who visits Blackjacktournaments.com knows about Smith's quandary of Doubling for Less when trailing as in Example(1) below shows. Infinite Deck...................Last Hand..................Min Bet = 5.....................Max Bet = 500 Player..........BkRoll...................Bet...................Cards BR2...............1000...............500 + 250..............2+3 BR1................1195.....................500.................10+10 Dealer....................................................................7 BR2 bets the max 500 and BR1 correlates by a matching bet of 500. BR2 now doubles for 250. The BR2 double is right in the middle of a double for 0 and a double for 500. The BR2 double cannot be correlated by BR1 by splitting since he cannot split for less. Actually any BR2 double of 200 up to 300 will put BR1 in the quandary. If BR1 stands he gives up the high to BR2. If BR1 splits he gives up the low to BR2. What should BR1 do? Below are the probabilities of BR1 winning the session by the simple but reasonably good strategy of splitting and playing not to bust against various dealer up cards. Also given are the dealer bust probability with the various dealer up cards. Dealer Up Card.............................................7..........................8......................10 Dealer Bust.............................................. .2623.................. .2447............... .2298 Prob of BR1 winning by splitting and playing not to bust............ .8004................ .. .7818............... .7492 The best way for BR1 in Example(1) to play against the quandary with a dealer 7 up card is figured as follows: The probability of BR1 winning by standing is BR1 needs the Dealer to make a hand. Prob(Make a Hand) = 1 - Prob(Bust) = 1 - .2623 = .7377. Since .7377 for standing is less than .8004 for splitting and playing not to bust; then splitting and playing not to bust is the better way to go. Instead of a dealer 7 up card in Example(1), what is BR1's best line of play against an 8 or 10 dealer up card and each of their probabilities? ..............................................................BlueLight as follows:

After 2 weeks nobody has tried to solve the teasers when the dealer has an 8 or 10 up. Player................BkRoll..........................Bet..............................Cards BR2....................1000......................500 + 250..........................2+3...............Doubles BR1.....................1195............................500.............................10+10..............Action? Dealer............................................................................................8 To solve the problem for BR1 standing when the dealer has an 8 upcard you need to know the probability or percent chance the dealer will make a hand. When the dealer has an 8 upcard he has the following total outcomes (that can also be found in other sources) : Dealer makes...................................probability...............................% chance ........21................................................ .0694 ...................................... 6.94 .......20................................................ .0694 ...................................... 6.94 .......19................................................. .1286 ..................................... 12.86 .......18................................................ .3593 ..................................... 35.93 .......17................................................ .1286 ...................................... 12.86 .....Bust.............................................. .2447 ...................................... 24.47 ..............................................Sum = 1.0000.........................Sum = 100.00 % Note that to account for all the possible outcomes the probabilities must add up to 1.0000 or the percent chances must add up to 100%. (I have seen some tables[but not at blackjacktournaments.com] where the sums did not add up to 1 or 100%.) We could add the chances of the dealer making a hand, by adding the chance of 21, the chance of 20, the chance of 19, the chance of 18 and the chance of 17. However by simply subtracting the chance of the of the dealer busting from 100% (or subtracting the probability of busting from 1) we get the total of the made hands probabilities. Dealer makes a hand = 1.0000 - dealer bust = 1.0000 - .2447 = .7553 for the situation in example(1) when the dealer upcard is an 8. (From the table given above for splitting) BR1 Splitting and playing not to bust yields .7818 .......................................BR1 standing yields .7553 Therefore BR1 splitting and playing not to bust is the better way to go. When the dealer upcard is a 10: Player................BkRoll..........................Bet..............................Cards BR2....................1000......................500 + 250..........................2+3...............Doubles BR1.....................1195............................500.............................10+10..............Action? Dealer...........................................................................................10 Here the dealer with a 10 up has a better chance to make a hand. BR1 standing with 10+10 is better than splitting and playing not to bust. Splitting and playing not to bust above was .7492 For BR1 standing, BR1 needs the dealer to make a hand. The probability of the dealer making a hand with a 10 upcard is : 1 - Prob(Bust) = 1 - .2298 = .7702 BR1 splitting = .7492 BR1 stands... = .7702 Therefore when the dealer has a 10 upcard then BR1 standing is the better way to go. ***However the teaser asked for the best line of play. Once it has been determined that splitting is not so good when the dealer has a 1o upcard, BR1 has a simple line of play which is easy to calculate that squeezes out and additional 1.77% chance (or 0.177 probability) over standing. What is that BR1 line of play. .......................................................................BlueLight

The rule of thumb, then, is split against dealer 9 and below, stand against dealer T and above. You're playing two hands each with a starting hand total of 10. So naturally it seems you should be the favorite in each hand against dealer upcards of 9 and below, but not so much against a dealer T or A. It seems like common sense, which makes it exceptional in the world of blackjack tournaments!

The best line of play for BR1 : Player..........BkRoll................Bet....................Cards BR2..............1000............500 + 250..............2+3..........Doubled BR1...............1195..................500..................10+10........Action? Dealer................................................................10 BR1 splitting and playing not to bust yielded... .7492 probability BR1 standing yielded......................................... .7702 probability However the best line of play for BR1 is by DOUBLING ON HIS 10+10!! It's by doubling for less by making a correlating double of 250. This way BR1 keeps the low and has a chance to improve to a lock if BR1 is delt the A. BR1 doubling for less (keeping the low) the worst that can happen is that BR1 busts. However if the dealer makes a hand (a .7702 chance) then wins the same as before. BR1 loses nothing by doubling (for less) and busting. BR1 will get a bust card 12/13 of the time. BR1 will get an A 1/13 of the time BR1 doubles and busts = (12/13) x (0.7702) = .7110 BR1 doubles and gets A = (1/13) x (1.0000) = .0769 ...................................................................Sum .7879........this gains .0177 over standing (.7702) The two starting low cards (2+3) for BR2 were not just two randomly chosen low cards. When BR2 doubles, regardless what card BR2 is delt, BR2 will have an unmade hand. It is necessary that BR2 ends up with total less than 17 for the double to be worth while. If BR2 had started with (3+4) and doubled and was delt a 10 for 17 then a dealer 17 (after a BR1 bust) would push for BR2 (in stead of losing) and BR2 would stay at 1000 (instead of dropping to 250) while BR1 would drop to 445. This possibility would more than offset the gain BR1 had when doubling (for less) when BR2 had an unmade hand. ......................................................................BlueLight

10+10 vs 15 or 17 The table below shows your chances when playing 10+10 vs a stiff or 17 The figures in bold show the best line of play. BR2.....1000.......500+250........2+3.......DD....10 = 15 BR1......1195.............500.........10+10......Action? Dealer......................................various up cards Up..............2.............3.............4.............5.............6.............7............8............9...........10............A Std........ .6433 .... .6233 .... .6029 .... .5823 .... .5605 .... .7377 .... .7553 .... .7716 .... .7702 .... .7993 Dbl....... .6707 .... .6523 .... .6335 .... .6145 .... .5943 .... .7579 .... .7741 .... .7891 . .7879 ... .8148 Splt...... .7641 .. .7712 .... .7786 .. . 7861 .. .7934 .. .8004 .. .7818 .... .7577 .... .7492 .... .7238 BR2......1000.......500+250......3+4.......DD....10 = 17 BR1.......1195............500.........10+10.....Action? Up..............2.............3.............4............5..............6............7..............8............9...........10...........A Std....... .6433 .... .6233 .... .6029 .... .5823 .... .5605 .... .7377 .... .7553 .... .7716 .. .7702 .. .7993 Dbl...... .5506 .... .5357 ..... .5205 .... .5052 .... .4883 .... .4176 .... .6554 ... .6784 .... .6765 .... .7381 Splt..... .7564 .. .7638 .... .7714 .. .7791 ... .7866 .. .7786 ... .7741 .. .7506 .... .7421 .... .7189 Note that when BR2 has 17; then when BR1 stands, BR1's chance of winning are the same as if BR2 had a stiff. Here however BR1 cannot improve by doubling for less; it only works against a BR2 stiff. I got the idea for this teaser when I saw the following come up in a tournament. BR2.......2000......500+500.........2+3.......DD.....10 = 15 BR1.......2260...........500............10+9.....stood? = 19 Fortunately for BR1 the dealer made a hand and he (BR1) won. BR1 would gain a little extra by doubling. The teaser here is what if BR2 had doubled for less (500+250)? Then what should BR1 do? Full double? Double for less? Or? ...........................................BlueLight BR1's best line of play?