Line Work

A problem proposed by C. Gebhardt in the Fall 1966 issue of Pi Mu Epsilon Journal:

A particular set of dominoes has 21 tiles: (1, 1), (1, 2), … (1, 6), (2, 2), … (6,6). Is it possible to lay all 21 tiles in a line so that each adjacent pair of tile ends matches (that is, each 1 abuts a 1, and so on)?

Click for Answer

A Bureaucracy Maze

At a Mensa gathering in 2003, Robert Abbott tried out a new type of maze — five bureaucrats sit at desks, and solvers carry forms among them:

When you enter the maze you are given a form that says, ‘Take this to the desk labeled Human Resources.’ You look for the desk with the nameplate Human Resources, you hand in your form to the bureaucrat at that desk, and he gives you another form. This one says, ‘Take this form to Information Management or Marketing.’ Hmm, there is now a choice. Let’s say you decide to go to Information Management. You hand in your form and receive one that says, ‘Take this form to Employee Benefits or Marketing.’ You decide on Employee Benefits where you receive a form saying, ‘Take this form to Corporate Compliance or Human Resources.’

Of the 30 participants, half gave up fairly soon, but the rest kept going until they’d solved it, taking 45 minutes on average. Here’s an online version with four desks, and here’s a fuller description of the project and its variants, including a Kafkaesque 2005 version by Wei-Hwa Huang in which the participants don’t know they’re in a maze.

More of Abbott’s logic mazes.

Black and White

abdurahmanovic puzzle

This puzzle, by F. Abdurahmanovic, won first prize in a 1959 Yugoslav tourney. It’s a helpmate — how can Black, moving first, cooperate with White to get himself checkmated in two moves?

Click for Answer

The Four Points, Two Distances Problem

winkler distances problem

Alex Bellos set a pleasingly simple puzzle in Monday’s Guardian: How many ways are there to arrange four points in the plane so that only two distances occur between any two points? He gives one solution, which helps to illustrate the problem: In a square, any two vertices are separated by either the length of a side or the length of a diagonal — no matter which two points are chosen, the distance between them will be one of two values. Besides the square, how many other configurations have this property?

The puzzle comes originally from Dartmouth mathematician Peter Winkler, who writes, “Nearly everyone misses at least one [solution], and for each possible solution, it’s been missed by at least one person.”

The answer is here.

Black and White

laws chess problem

“A fairly good two-mover” from Benjamin Glover Laws’ The Two-Move Chess Problem, 1890. What’s the key move?

Click for Answer


Is 94,271,013 the sum of 12 consecutive integers?

Click for Answer