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Problem of the Week #148: Solution
07-30-2012
, 10:58 PM
Problem of the Week #148: Solution
Cash game, Black owns a 2-cube.
Black to play 1-1. Where does the blot belong?
Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.
This is a cute little problem because Black can’t get safe, but he can leave his blot on any of five different points. As he moves closer to White’s midpoint, he lowers the chance that he gets hit, but he also lowers the chance that White can leave a shot. So where does the checker belong – close up or farther back?
Since the home boards are closed, let’s stipulate that Black will lose if he gets hit, but win if he hits. We’ll further stipulate that if no hitting occurs, Black is slightly better off the closer his checker is to home, but that effect is tiny compared to the importance of hitting or being hit. The net result is that we want to pick the play that minimizes the net number of times that Black is hit.
The answer should be pretty clear intuitively, but the calculation is easy so here’s a table that shows what’s going on in detail:
The first column shows where Black’s checker ends up after he picks a play, while the second column show how many times Black gets hit (out of 36 rolls). Playing 18/14 leaves the fewest shots (12/36), while leaving the blot on the 18-point leaves the most shots (17/36).
Column 3 shows the number of White rolls that leave a shot after each play. If Black plays 18/14, for instance, White can only leave a shot if he rolls 6-1. But if Black plays 18/17, White can leave a shot with 6-1, 6-2, 6-3, or 6-4. Note that if Black stays where he is, White can’t leave a shot. Every number either hits or plays safe.
Column 4 shows the number of times (out of the original 36) that Black hits the shots from column 3. If he plays 18/14 and White rolls 6-1, for instance, Black will hit (2 * 11/36), or 22/36, or 0.6 out of the original 36.
Column 5 is just (2) – (4), that is, the number of times Black gets hit minus the number of times he hits White. A smaller number is a better number.
Looking at the table shows that 18/14 is the best play. Minimizing the immediate shots is more important than picking up shots on the following turn, since then Black has to both get a shot and hit it.
The general point is simply that minimizing the bad things that can happen on the next turn is more important than minimizing a parlay, where multiple things have to occur. The probability of a parlay is the probability of all the individual events multiplied together, and that tends to be a very small number.
Solution: 18/14
Cash game, Black owns a 2-cube.
Black to play 1-1. Where does the blot belong?
Note: All ‘cash game’ problems assume the Jacoby Rule is in effect. That is, you can’t win a gammon unless the cube has been turned.
This is a cute little problem because Black can’t get safe, but he can leave his blot on any of five different points. As he moves closer to White’s midpoint, he lowers the chance that he gets hit, but he also lowers the chance that White can leave a shot. So where does the checker belong – close up or farther back?
Since the home boards are closed, let’s stipulate that Black will lose if he gets hit, but win if he hits. We’ll further stipulate that if no hitting occurs, Black is slightly better off the closer his checker is to home, but that effect is tiny compared to the importance of hitting or being hit. The net result is that we want to pick the play that minimizes the net number of times that Black is hit.
The answer should be pretty clear intuitively, but the calculation is easy so here’s a table that shows what’s going on in detail:
The first column shows where Black’s checker ends up after he picks a play, while the second column show how many times Black gets hit (out of 36 rolls). Playing 18/14 leaves the fewest shots (12/36), while leaving the blot on the 18-point leaves the most shots (17/36).
Column 3 shows the number of White rolls that leave a shot after each play. If Black plays 18/14, for instance, White can only leave a shot if he rolls 6-1. But if Black plays 18/17, White can leave a shot with 6-1, 6-2, 6-3, or 6-4. Note that if Black stays where he is, White can’t leave a shot. Every number either hits or plays safe.
Column 4 shows the number of times (out of the original 36) that Black hits the shots from column 3. If he plays 18/14 and White rolls 6-1, for instance, Black will hit (2 * 11/36), or 22/36, or 0.6 out of the original 36.
Column 5 is just (2) – (4), that is, the number of times Black gets hit minus the number of times he hits White. A smaller number is a better number.
Looking at the table shows that 18/14 is the best play. Minimizing the immediate shots is more important than picking up shots on the following turn, since then Black has to both get a shot and hit it.
The general point is simply that minimizing the bad things that can happen on the next turn is more important than minimizing a parlay, where multiple things have to occur. The probability of a parlay is the probability of all the individual events multiplied together, and that tends to be a very small number.
Solution: 18/14
