Tantalizer 1: Publish or perish
From New Scientist #550, 22nd June 1967 [link]
Kappa, Lambda, Mu and Omicron are at present uneasily seated in the Warden’s study at Jude’s College, awaiting summonses from the committee which will appoint one of them to the vacant Fellowship in Greek Literature. Each is hugging his only published work and each suspects that the post will go to the author of the longest, irrespective of all possible merit.
From their stilted but cunning conversation, the following facts have so far emerged:
Each book has a whole number of pages over 100.
Only Lambda’s book and Mu’s book have the same number of pages.
The total number of pages in all four books is 500.
Mu then asked Omicron whether the number of his (Omicron’s) pages was a perfect square. From Omicron’s answer Mu and Kappa made silent and independent deductions with impeccable logic. Mu deduced that Omicron’s book was the longest. And Kappa, who was not a perfect square, deduced that Omicron’s answer was not the truth.
How many pages are there in each man’s book?
This was the first Tantalizer puzzle published in New Scientist. It was accompanied by the following introduction:
This is the first of a series of logical puzzles compiled by Martin Hollis. No mathematical knowledge is required for their solution. A new puzzle will appear each week, and the answer will be printed in the following week’s issue.
M knows his own number, and from O’s claim he can deduce that O has the largest number.
K knows his own number, and can deduce the O’s claim is false.
And from the intersection of the two sets of scenarios we can determine what the actual numbers must be.
This Python program runs in 66ms. (Internal runtime is 5.2ms).
Run: [ @replit ]
Solution: Page counts are: Kappa = 156; Lambda = 121; Mu = 121; Omicron = 102.
The only way M can deduce that O has the largest number of pages, is if M has 121, and O claims his number is a perfect square. Then M deduces O has 144 (and K has 114). However M has been misled by O’s claim.
There are many values that K can have, and deduce O’s claim must be false. Only two of them coincide with “M = 121 and O is not a perfect square”.
But we are told K is not a perfect square, so we can eliminate the second of these, leaving a unique solution.