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The challenge of this puzzle is to deduce enough of the rules of the card game Cleaving Hearts to assign points correctly to the winners of each trick. Most — although not all— of the rules of this game (which was invented for this puzzle) can be deduced from the example game fragment that forms the puzzle. Here are the rules in full:
A key observation in this puzzle is that, as the game progresses, the scores of players can be used to spell words. (Realising this also helps clarify some uncertainties about the rules, and smooth over bits that are unclear.) Player 3 is the first to win one heart, achieving a score of one, which can be interpreted as the letter A. Player 1 then scores two hearts, achieving a score of two, which can be interpreted as the letter B. Following this process in round 1 results in the letters ABRACADABRAH. Note the H at the end.
It is the winner's scores that determines the letters. In some tricks there are multiple winners and multiple letters may be extracted from such tricks. In the first round, two winners produce the letters CA in trick 9 by scoring one point each, and then three winners produce DAB in trick 10, again by scoring one point each. These tricks should provide some insight into the rule of dividing point-scoring cards equally.
In round 2, the scores continue to increase, and the first letters obtained are OCU. These letters follow the H at the end of the first round, and it may be surmised that the word HOCUS is intended. However, the S is not so easy to extract. It is here, in tricks 8 and 9, that the jackpotting rule must be deduced to obtain an S. Doing so produces the full set of round 2 letters: OCUSPREST.
Round 3 produces the letter O (completing the word PRESTO) and this is followed by TONTUS. Jackpotting in tricks 12 and 13 produces no new letters, and the rule about not jackpotting in the final trick must be deduced to avoid changing the scores further. Note: TONTUS is an obscure "magical word", similar to ABRACADABRA, but none of these words are the solution to the puzzle. (The invaluable Cecil Adams provides details about the origins of these words.)
The full worked solution is below. Leads are outlined and winners have blue backgrounds. Running totals of player scores are written in blue, with the corresponding letters next to them.
In the end, the final scores are tallied and the corresponding letters can be read left to right, spelling the answer to this puzzle: SURTOUTS. (A surtout is a type of coat, which might feasibly be worn by some magicians.)
Can a puzzle teach a game? Can a game also be a puzzle? I decided to find out. The central idea for this game was that there could be multiple winners of a trick, and hence the lead could become split. There would be a corresponding mechanism for merging back to fewer leads, by having fewer winners. This repeated splitting and merging of the lead was the source of the name Cleaving Hearts since in English 'to cleave' can mean to split but can also mean to join.
Although I've given many rules above, not all of them can be (or need be) deduced from the sample game. For instance, knowledge of The Pass is not explicitly required to solve the puzzle. It may be inferred from trick 5 of round 1 when player 2 plays a king of spades while there is still one card remaining to be played in the trick, tempting fate by potentially allowing the next player to drop a queen of spades. This hazard would be unlikely if player 2 had passed cards to player 3 before the first trick, and thus knows that no queen of spades was passed to them.
Similarly, the simultaneous nature of play when there are multiple leads might not be strictly required to solve the puzzle, but may nevertheless be inferred by observing how suits are followed and which players duck just under the currently winning cards. Trick 11 of Round 3 shows players 4 and 6 leading spades and hearts, and the subsequent players 5 and 7 both follow hearts, showing that player 5 is aware of player 6's lead and need not "follow from the left". A similar situation occurs in trick 4 of round 2, where players 6 and 8 lead, then players 7 and 1 go low, but once player 2 goes high the remaining players duck just under player 2's card. This illustrates what happens when two simultaneous branches reduce to a single line of play.
The rules concerning Shooting the Moon were given in the summary above, but no player attempts this in the puzzle, and it is not necessary to know this rule to solve the puzzle. The rule is stated for completeness. Whether a player had successfully or unsuccessfully attempted this goal, it would have required a different formulation of the puzzle, since a change of up to 52 points in score would not lend itself so easily to a direct alphabetic mapping from scores to words. So, unfortunately, this rule was not incorporated into the puzzle.
The Jackpot rules were perhaps the most difficult to discover. The embedded magical words were both a mechanism to cross-check the deduced rules and an aid to discover new rules. The puzzle was extensively reworked after initial test-solves, moving the first instance of a jackpot from trick 2 of round 2 down to trick 8, to coincide with the final letter of HOCUS.
The distribution of points in cases of multiple winners was another challenging aspect of this puzzle. The need to think in terms of scoring by physically handing cards to players was one of the intuitive leaps required to solve the puzzle. Many test-solvers considered multiple ways of dividing points between winners, particularly when a queen of spades was present in the trick. Again, the embedded magical words could be used to cross-check these options (or, in at least one test-solver's case, to bypass a full understanding of this process).
When constructing this puzzle, it was important to find words with letters that were small increments on earlier words (up to a difference of 8, formed using hearts cards), or else large increments (a difference of 13 or more, formed using a queen of spades plus one or more hearts cards). Abracadabra was an ideal first word, and had a suitable thematic link with magical tricks. If a difference in letters was between 9 and 12, this required jackpotting to spill some points over to a later trick. This occurs in tricks 8 and 9 of round 3. Jackpotting was also useful for eliminating some points at the end of round 3, to make it more difficult to reverse engineer the secret, which would otherwise simply be the sum of three rounds each containing 52 points.
During test solving, while the puzzle was being considered for day 3, an idea was floated of hand writing "How magical" under the puzzle to hint at Abracadabra and provide an easier way in. After some discussion, this was rejected as there was a need for truly hard puzzles for day 5.
Also during test solving, several solvers reinvented one of the rules of Cancellation Hearts, where equal highest ranked cards cancel each other out. That variant of Hearts also has a form of jackpotting since it is possible for an entire trick to become cancelled if there are multiple ties. The point-scoring cards would still count even if they are part of a tie, but in a jackpot situation they'd spill over to the next trick. Cancellation Hearts also removes a two of clubs in order to ensure a single lead for each trick, and the cancellation mechanism attempts to ensure a single winner, thus making the game quite similar to Hearts.
By constrast, Cleaving Hearts embraces the notions of multiple leads and multiple winners. The game can be adapted for fewer than 8 players through the addition of Jokers as non-winning cards.
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Alternatively, some low-ranked cards may be removed to balance the hands, although both twos of clubs should be retained to ensure the possibility of an opening split lead. For six players, in particular, a sensible alternative would be to remove both twos of diamonds instead of adding Jokers, dealing seventeen cards to each player.