The Grapevine

A problem from the British Columbia Colleges Senior High School Mathematics Contest, 2000:

Not all of the nine members on the student council are on speaking terms. This table shows their relationships — 1 means two members are speaking to each other, and 0 means they’re not:

  A B C D E F G H I
A - 0 0 1 0 0 1 0 0
B 0 - 1 1 1 1 1 1 1
C 0 1 - 0 0 0 1 1 0
D 1 1 0 - 1 0 1 0 1
E 0 1 0 1 - 0 1 0 0
F 0 1 0 0 0 - 0 0 1
G 1 1 1 1 1 0 - 0 0
H 0 1 1 0 0 0 0 - 0
I 0 1 0 1 0 1 0 0 -

Recently councilor A started a rumor, and it was heard by each councilor once and only once. Each councilor heard it from, and passed it to, another councilor with whom she was on speaking terms. If we count councilor A as zero, then councilor E was the eighth and last councilor to hear the rumor. Who was the fourth?

Click for Answer

Murder Most Classificatory

In Anthony Boucher’s short story “QL 696.C9,” a librarian is found dead at her desk. She has been shot, and apparently spent her last moments typing the Library of Congress card catalog number that gives the story its title. The killer evidently saw nothing incriminating in this and so left it alone. The investigators have narrowed the list of suspects to junior librarian Stella Swift, children’s librarian Cora Jarvis, library patron James Stickney, and high school teacher Norbert Utter. Who did it?

Click for Answer


A problem from the British Columbia Colleges Senior High School Contest for 2000:

If I place a 6 cm × 6 cm square on a triangle, I can cover up to 60% of the triangle. If I place the triangle on the square, I can cover up to 2/3 of the square. What is the area, in cm2, of the triangle?

(a) 22 4/5
(b) 24
(c) 36
(d) 40
(e) 60

Click for Answer

Black and White

shinkman chess problem

“We now give what is acknowledged to be the finest two-move problem extant,” wrote J.H. Blackburne in the Strand in 1908. “It is by the American expert, W.A. Shinkman, and is also claimed by G.E. Carpenter, a fellow countryman of his. Here we have not only a difficult key-move, but also beauty of theme and artistic construction, the three essential qualities necessary to a perfect problem.”

White to mate in two moves.

Click for Answer

The Fifth Element

pmej matrix puzzle

Charles W. Trigg offered this puzzle in the Fall 1977 issue of Pi Mu Epsilon Journal (PDF):

This square array contains the first 25 positive integers. Choose five, no two from the same row or column, so that the largest of the five elements is as small as possible, and justify your choice.

Click for Answer

The Impossible Puzzle

Dutch mathematician Hans Freudenthal proposed this puzzle in 1969 — at first it appears impossible because so little information is given.

X and Y are two different whole numbers greater than 1. Y is greater than X, and their sum is no greater than 100. S and P are two logicians; S knows the sum X + Y, and P knows the product X × Y. S and P both reason perfectly, and both know everything I’ve just told you.

  • S says, “P does not know X and Y.”
  • P says, “Now I know X and Y.”
  • S says, “Now I also know X and Y.”

What are X and Y?

Click for Answer