Hearts: Answer to question 15

Q: I walked in on a game just after the last hand had been scored. When I saw the scores, which were in arithmetic progression, I knew that someone had Shot the Moon in the last hand - and I knew who it was. What could the winner's score be?

A: 5 or 92 or 97

I liked these solutions too much to artificially exclude any, just to get a unique answer. Toby Gottfried and Neil Smith alerted me to options (b) and (c).

If I knew that someone had Shot the Moon in the last hand, then we have three (not necessarily mutually exclusive) possibilities:

(a) The losing score could have been 125, as it is a score not reachable other than by someone Shooting the Moon. There are three instances of four positive integers

• in arithmetic progression with positive increment
• which have highest number 125  (as per question 1)
• which have lowest number less than 103 (as per question 2)
• for which the sum is a multiple of 26

From the complete list of end scores in arithmetic progression, these are

    83  97 111 125
    44  71  98 125
     5  45  85 125

Only with the last combination of scores can I determine that the player with a score of only 5 must uniquely have been the player to have Shot the Moon, as otherwise that player's score would be 26 or more.

(b) If no one shoots the moon, then only 26 points are scored on the last hand, at most one player can score 13 or more points, except for the case 0,0,13,13.

If one player scores at least 99 + 13 = 112 and another scores at least 113 (eliminating the special case above), we can conclude that there was a Shoot the Moon on the last hand.

In such a case, if the other two players score under 100, we would not know which of them had Shot, so we are reduced to cases where, in addition, there is a third score over 99.

The following combination of scores satisfies these conditions.

    97 106 115 124

    92 100 108 116
    97 106 115 124 (as before)

where again the winner must have shot the moon to end up with this final score as 

• the total of points above 99 more than 26 (ie more than a moonless last hand) and
• all other (non-winning) players have scores of 100 or more (so could not have Shot the Moon).