“Professor! You missed an incredible game at Jourdan’s last night. In the first two rounds, Grover and I were doubled in spades on every board! The way Grover bids, that’s usually our opponents’ best strategy, but this time it backfired. Amazingly, we made every one! Plus over four thousand on six boards, 4360 to be exact. Have you ever seen anything like that?”
“Can’t say that I have, Timothy. I’ve certainly been plus more than that, but six consecutive contracts in the same suit is remarkable. Did Jourdan’s have a big crowd?”
“No, just nine tables but a perfect Mitchell. Even more bizarre is that we played every spade contract from one to six, and each score was higher than the previous one — like a progressive blitzkrieg.”
“That is extraordinary; but even so, I’m betting you and Grover didn’t win.”
“[Sigh] Good bet. After the opening streak, every hand went sour and we barely broke average.”
“Predictable. Accelerated collisions cause supercharged pions to decay rapidly into gamma rays, which according to my theory cause boson deflection to become unstable. This transforms the higher-massed spade quarks into antimatter, rendering the bulk of those contracts unplayable.”
“You took the words right out of my mouth, or at least ‘antimatter’ describes Grover pretty well.”
Clarifications: Each spade contract (1 to 6 ) was doubled but not redoubled. Overtricks may or may not have been made (no restrictions). Only the six scores (not necessarily the contract levels) were in ascending order.
Test your power of deduction, or make your best guess!
What direction were Timothy and Grover?North-South East-West
What pair number were Timothy and Grover?1 2 3 4 5 6 7 8 9
Tiebreaker: Based on the stated occurrence and movement, what isthe maximum score any pair could achieve on the six boards?5860 6460 6910 7410 7560 9060
To see how you did click
In August 2016 this puzzle was presented as a contest, inviting anyone who wished to submit a solution. Participation was the third most in this series with 59 persons giving it a try. Only the 20 listed below correctly deduced the pair number of Timothy and Grover.
Trending on the recent Olympics, our gold medal goes to Nicholas Greer, who was the first of 10 to be exact on the tiebreaker. This is Nick’s first win, though it’s hardly a surprise with his high finish almost every month, including a second, two thirds, a fourth, two fifths, two sixths and two sevenths. The silver medal goes to Dan Gheorghiu, our second most prolific winner, and the bronze to our all-time leader, Timothy “sans Grover” Broeken. As they take their positions on the podium, all please rise for “God Save the Queen.”
The key to solving this puzzle is to consider the vulnerability sequence faced by each pair in their first six boards. In a Mitchell movement, North-South pairs play board numbers in order, while East-West pairs skip a set each round. The following table shows the first six boards played by each pair and the corresponding vulnerability of their side. (Opposing vulnerability doesn’t matter with all contracts making.)
The six grayed pairs are unlikely candidates, because their vulnerability sequence is not unique. For example, if Pair 8 NS were able to score 4360, so also could Pair 9 NS; hence the puzzle would have two solutions.
Next organize all the possible results, as in the table below. Each cell shows the two possible scores (nonvul and vul) for that result. Note that 1 × making one is a special case, yielding the same score vulnerable or not.
One observation is noteworthy: The 10s digit of all scores in each contract from 1 × to 5 × is the same, so five of the six scores must end in 60, 70, 30, 90 and 50, which sum to zero (modulo 100). Therefore, to reach a total ending in 60, 6 × must have been made vulnerable for 1660 or 1860.
Mike Wenble: The aggregate score constrains the 6 × contract to be vulnerable. Lots of valid combinations sum to 4360, but most require three or four consecutive nonvul hands which is not possible.
Tom Slater: The [60 ending] gives us the vulnerability and position of the small slam, then it’s just a matter of trying some combinations.
Richard Morse: The slam has to be vulnerable to have an answer ending in 60, and the rest is constrained by the rather low aggregate.
Determining all the ways to get 4360 would be tedious by hand, but a breeze by computer, which found 94 different ways. The actual scores, however, don’t matter. All that matters is the vulnerability sequence of the six contracts when the scores are placed in ascending order. In that regard there are just six cases:
Case 1 can be eliminated since it has only one vulnerable contract, while the minimum any pair faces is two. All that remains is to compare Cases 2-6 with the vulnerability sequence for each pair. Lo and behold, there is one match: Case 6 = Pair 3 NS. Hello, Timothy and Grover! The three ways to score 4360 are listed below, each having the same first and last result. Tinting shows the vulnerability.
Sherman Yuen: Timothy and Grover were Pair 3 NS with 1 ×= 160, 3 ×= 530, 4 ×= 590, 2 ×= 670, 5 ×+1 750 and 6 ×= 1660.
Nicholas Greer: The double of 5 on Board 11 may have been a stripe-tailed ape double, but with the theme of this puzzle I’ll assume not, [lest] it have been the opponents’ best strategy.
Mike Frentz: Last deal must be 6 ×= 1660 (vul)… consistent with seating for NS(3,4,5,6) and EW(1,8,9). Minimum remaining is 2400 (160+470+530+590+650), leaving 300 in vulnerability-overtrick bonuses. Board 1 must be 1 ×, as minimum score for others is 470… At most one of boards 2-5 can be vulnerable, leaving only NS(3,5) and EW(8). Board 1 score must be 160, as 260 makes it impossible for the vulnerable board to come in the correct place. Only solution is NS Pair 3.
Dan Baker: Trick scores, doubled insults, one partscore bonus, five nonvul game bonuses, and nonvul slam bonus add to 3610. Only way to reach a total ending in 60 is to make the slam vulnerable, which gets them to 4060. [Dan similarly deduces Pair 3 NS].
The 4360 scored by Timothy and Grover was a modest total for the occurrence (minimum possible is 3910). For the tiebreaker, the goal was to find the maximum. Choosing the best score for each contract yields a hefty sum of 9060, but it doesn’t wash; all those scores are vulnerable, and no pair is vulnerable on all six boards. The most vulnerable deals of any pair is four, specifically 4 NS, 7 EW and 9 EW.
Since scores must be ascending, it is better to have vulnerable deals toward the end of the six boards. Pair 4 NS (V N V V N V) seems the best candidate, since pairs 7 and 9 EW each have two vulnerable deals at the start. Working from the high end, a little tinkering shows that 1670 must be the top score so that 1310 (best nonvul by far) can be next, followed by 1250 and 1190 (next highest vuls) then 970 and 960 (best fitting nonvul and vul) for a total of 7210. But can we do better?
Dan Gheorghiu: The Professor’s boson deflection theory was in effect, as I could not find a suitable way to [formulate] the optimal score. Therefore, I turned to intuition and trial and error. Including the highest scores on vulnerable boards and the highest nonvulnerable score (1310) became maximum at boards 16-21.
Yes, even better than being vulnerable four times, is to be vulnerable on the last two deals. The only pair to claim that is 6 NS (N N V N V V) and they alone can achieve the maximum possible aggregate score of 7410. This can be done in three ways as shown below. Note that each result except one (1 ×+5) contains the most possible overtricks (making seven) so it’s hardly realistic, but then, puzzles seldom are.
Tom Slater: Only Pair 6 NS can get the greatest score, being the only pair to finish N-V-V.
Nicholas Greer: The maximum score is achieved by Pair 6 NS making 13 tricks every time in 1, 4, 5, 6, 3 then 2 ×.
Sherman Yuen: Pair 6 NS can score 7410 with 5 ×+2 850, 4 ×+3 890, 1 ×+5 1160, 6 ×+1 1310, 3 ×+4 1530 and 2 ×+5 1670.
Ivan Loy: I will never double another spade contract! Why take the risk when 4 can be cold with zero HCP in both hands?
Wayne Somerville: Everyone at the club seems concerned that I supply the scores for making doubled contracts too fast.
Charles Blair: “If you set every contract you double, …”
© 2016 Richard Pavlicek