Empty Vessel

A puzzle by Soviet science writer Yakov Perelman: A bottle full of gasoline has a mass of 1,000 grams. The same bottle filled with acid has a mass of 1,600 grams. The acid is twice as dense as the gasoline. What’s the mass of the bottle?

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Win Count

Imagine a game of tic-tac-toe (noughts and crosses) played in three dimensions in an 8×8×8 cube. A player wins by marking some straight line of eight cells through the large cube. How many such winning lines are there?

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“Three Threes Are Ten”

This little trick often puzzles many:–

Place three matches, coins, or other articles on the table, and by picking each one up and placing it back three times, counting each time to finish with number 10, instead of 9. Pick up the first match and return it to the table saying 1; the same with the second and third, saying 2 and 3; repeat this counting 4; but the fifth match must be held in the hand, saying at the time it is picked up, 5; the other two are also picked up and held in hand, making 6 and 7; the three matches are then returned to the table as 8, 9, and 10. If done quickly few are able to see through it.

— John Scott, The Puzzle King, 1899

04/20/2024 Reader Vladamir Tsepis adds, “This reminds me of the way to convince children you have 11 fingers. Start by showing your left hand splayed, curl down the thumb and index finger counting ‘one, two…’, then of the remaining say ‘let’s skip these three’. Move to your right hand, bend each finger in turn as you count ‘four, five, six, seven, eight…’. Return to the left hand counting off the three we skipped ‘nine, ten, eleven.'”


From The Book of 500 Curious Puzzles, 1859:

Following is the epitaph of Ellinor Bachellor, an old pie woman. How should we read it?

Bene A. Thin Thed Ustt HEMO. Uld yo
L.D.C. RUSTO! Fnel L.B.
Ach El Lor. Lat. ELY,
Wa. S. shove N. W. How — Ass! kill’d I. N. T. H.
Ear T. Sofp, I, Escu Star.
D. San D T Art. San D K. N E. W. E
Ver — Yus E. — Oft He ove N, W. Hens He
‘Dli V’DL. on geno
Ug H S hem A.D.E. he R. la Stp. Uf — fap
Uf. F. B Y he. R hu
S. Ban D. M.
Uch pra is ‘D. No. Wheres Hedot
HL. i. e. Tom. A kead I.R.T.P. Yein hop Esthathe
L L B. Era is ‘–D!

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Counting Up

A problem from Daniel J. Velleman and Stan Wagon’s excellent 2020 book Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles:

A square grid measures 999×999. Each square is either black or white. Each black square that’s not on the border of the grid has exactly five white squares among its eight immediate neighbors (those that adjoin it horizontally, vertically, or diagonally). Each white square that’s not on the border has exactly four black squares among its immediate neighbors. Of the 999 × 999 = 998001 squares in the grid, how many are black and how many white?

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A puzzle by Soviet science writer Yakov Perelman: Six carpenters and a cabinetmaker were hired to do a job. Each carpenter was paid 20 rubles, and the cabinetmaker was paid 3 rubles more than the average wage of the whole group. How much did the cabinetmaker make?

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Mass Transit

A problem from the October 1964 issue of Eureka, the journal of the Cambridge University Mathematical Society:

The planet Kophikkup is in the shape of a torus or ring-doughnut. There is a direct mono-rail line from each of the four space-ports to each of the major cities. No lines join or cross. What is the greatest possible number of major cities? Draw a diagram for this case.

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“The Twenty-Four Monks”


During the middle ages there existed a monastery, in which lived twenty-four monks, presided over by a blind abbot. The cells of the monastery were planned as shown in the accompanying figure, passages being arranged along two sides of each of the outer cells and all round the inner cell, in which the abbot took up his quarters. Three monks were allotted to each cell, making, of course, nine monks in each row of cells. The abbot, being lazy as well as blind, was very remiss in making his rounds, but provided he could count nine heads on each side of the monastery he retired into his own cloister, contented and satisfied that the monks were all within the building, and that no outsiders were keeping them company. The monks, however, taking advantage of their abbot’s blindness and remissness, conspired to deceive him, a portion of their number sometimes going out and at other times receiving friends in their cells. They accomplished their deception, and it never happened that strangers were admitted when monks were out, yet there never were more nor less than nine persons upon each side of the building. Their first deception consisted in four of their number going out, upon which four monks took possession of each of the cells numbered 1, 3, 6, and 8, one monk only being left in each of the other cells; nine monks being thus on each side of the building. Upon returning, the four monks brought in four friends, when it was necessary to arrange the twenty-eight persons, two in each of the cells 1, 3, 6, and 8, and five in each of the others; still nine heads only were to be counted in either row. Emboldened by success, eight outsiders were introduced, and the thirty-two persons now were arranged, one only in each of the cells 1, 3, 6, and 8, but seven in each of the other cells; again, according to the abbot’s system of counting, all was well. In the next endeavour, the strangers all went away and took six monks with them, leaving but eighteen at home to represent twenty-four; these eighteen placed themselves five in each of the cells 1 and 8 and four in each of the cells 3 and 6; the remaining cells were empty, but the cells on each side of the building still contained nine monks. On returning, the six truants each brought two friends to pass the night, and the thirty-six retired to rest, nine in each of the cells 2, 4, 5, and 7; the remainder were empty, and the abbot was quite satisfied that the monks were alone in the monastery.

Cassell’s Book of In-Door Amusements, Card Games and Fireside Fun, 1882

Target Practice


An arrow has a 1/4 chance of hitting its target. If four arrows are shot at one target, what’s the chance that the target will be hit?

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Here’s a checkerboard. Suppose we put a checker on each of the nine squares in the lower left corner. And suppose that any checker can move in any direction by jumping over an adjacent checker, provided that the square beyond it is vacant. Is there some combination of moves by which we can transfer the nine checkers to the nine squares in the upper left corner of the board?

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