E N I G M A

MURDER AT THE LIAR’S CLUB

by Mark.J.P. Wolf

There’s been a murder at the Liar’s Club in Merm, and as interim sheriff, you are called upon to investigate. Your deputy, Ned, did some preliminary interrogation, but has given up in frustration; members of the Liar’s Club always lie, and there are at least three members, and at least three non-members (truth-tellers), among the seven people involved (the six suspects and the murdered man). Everyone

involved knows who is a member and who isn’t, and even who the killer is, but no one wants to get involved, for fear of becoming the next victim. Ned’s first question to the six suspects was ‘Are you a member?’, to which each person answered ‘No.’ Finally in desperation, Ned took a statement from each suspect, leaving you to sort out the clues. Along with the statements is a written statement from Hart, the murder victim, which he wrote as he was dying. In his note, he accuses Archer of killing him; however, Hart may be a member, and thus a liar. Below is a summary of the six suspects statements, and Hart’s note which he wrote as he died. The case is definitely not suicide, and it is up to you to determine from the seven statements who is a member and who isn’t, and who the killer is.

Archer: ‘Davis is a member, and he would agree that Flint would agree that Hart and Edgar are both members.’

Brown: ‘Hart would agree that Clark would agree that Archer and Davis are both non-members.’

Clark: ‘Davis and Brown would agree that Edgar would agree that the murderer is not a member.’

Davis: ‘Hart’s membership status is the same as the murderer’s.’

Edgar: ‘Clark would agree that Brown and Hart would agree that Flint is a member.’

Flint: ‘Brown would agree that Archer and Clark do not have the same membership status.’

Hart: ‘Archer killed me!’

(Answers to appear in the next issue.)

See next page for answer to last issue’s puzzle.

Congratulations to Sasha Chislenko for being the first to solve last issue’s Enigma.

---EXTROPY #16 Q1 ‘96

48

SQUARED DEAL — SOLUTION

Republic	Side length (miles)	Land Area (square miles)
Zevo	2	4
Elga	4	16
Koid	6	36
Dorra	7	49
Rudra	8	64
Merm	9	81
Froll	11	121
Gom	15	225
Hort	16	256
Okell	17	289
Curro	18	324
Lenif	19	361
Slome	24	576
Telka	25	625
Phydra	27	729
Wintz	29	841
Jarp	33	1089
Nurin	35	1225
Bolta	37	1369
Voth	42	1764
Alto	50	2500
The USR	112	12544

Quadra, the capital city, lies in the republic of Merm.

The first step in solving the problem is to determine the sizes of the republics, and clue A. helps in finding the range of sizes. Since there are 21 squares of unequal size, the second largest square would have to be at least 20 miles wide; thus the maximum size of the largest square would be limited to 92 miles wide (otherwise they could not fit within the USR). From clue A., the largest square must be either 75 or 50 miles wide (25 is too small a size for the largest square, since the total for the 21 unequal squares comes to 1122 =12544 square miles). The smallest territory, then is either 2 or 3 miles wide.

When clue B. is taken into account with clue A., the size of the second largest square must be either 28, 35, 42, or 49 miles wide, and the third smallest square must be either 4, 5, 6, or 7 miles wide, as these are the only combinations possible which have values that fit inside the bounds set in the two clues, and the USR size of 112 miles square. If the largest square is 75, then only the {35, 5} combination will work for clue B., since the room left in the 112 x 112 square will not fit squares bigger than 37 alongside a 75 mile wide

square. If the 35 mile one is used, however, there will be a 2-mile gap, and since 3 is the smallest square, this combination will not work. Thus, the largest square is not 75, but 50, and the smallest square is 2 miles wide.

Using the rest of clue A., and knowing 50 to be the largest square, and 2 to be the smallest, we can determine the six smallest squares to be 2, 4, 6, 7, 8, and 9 (two combinations of six squares add up to 250, but only one satisfies Clue B.). Also, since we know 6 to be the third smallest, by clue B. that the second largest square is 42 miles wide.

Clue E. reveals that Nurin is 1225 square miles, and that Gom is 225 square miles. Since Bolta is larger than Nurin, and Gom is 225, Voth must be more than 1450 square miles, and thus must be either 40, 41, 42, or 50 miles wide (taking clue C. into account). The value 40, however, does not work, so Voth must be either 41, 42, or 50 miles wide, since combinations of squares are possible for all three values. Since there are only four squares larger than 225 by less than 150, we know (by the rest of clue E.) that 16, 17, 18, and 19 are side-lengths of four other republics.

Clue F. reveals Telka as the sum of

two squares; the smallest square which is the sum of two squares is 100, so Telka’s length must be at least 10; Telka, then, could be 10, 13, 15 (which can be eliminated because Gom is 15), 17, 20, or 25 miles wide. Numbers over 25 are too big, since Alto, which is four times the area of Telka, can be no bigger than 50. Thus, Alto must be 20, 26, 34, 40, or 50 miles wide. This also limits the possibilities for the Slome & Dorra combination to {5,12}, {6, 8}, {7, 24}, {8, 15}, and {12, 16}. But, by clue C., 5 is not a possibility for square width, and Gom is 15, so Slome & Dorra are either {6, 8}, {7, 24}, or {12, 16}. Telka is either 10, 20, or 25 miles wide, and Alto is either 20, 40, or 50 miles wide.

By clue G., we know Froll is larger than at least two other republics, so it must be 6 or more miles wide. Since Froll cannot be 4, Voth cannot be 41, ruling out one of the possibilities. Voth must be either 42 or 50. Given the possible values for Dorra (8, 16, and 24), we see that one of remaining possibilities, {Voth=50, Gom=15, Bolta + Dorra + Froll = 25 + 27 + 39}, does not work either, because Dorra can be none of those three values. Therefore Voth must be 42 miles wide, making Bolta= 37 miles wide, Froll= 11 miles wide, and Dorra= 7 miles wide. Taking clue F. into account, the only value possible for Slome which is a square when added to 49, is 24, so Slome= 24 miles wide, Telka= 25 miles wide, and Alto= 50 miles wide.

From the preceding steps, we know the values and names of nine of the repub-

Cont. on p.64

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EXTROPY #16 Q1 ‘96