# Tasks of a mad mathematician

Antón Chéjov published * Tasks of a mad mathematician * in 1882 under the pseudonym “Antosha Chejonté”. It is a hilarious parody of mathematical work through eight absurd questions posed by a "crazy" mathematician and which are reproduced below (see [1]):

I was chased by 30 dogs, of which 7 were white, 8 gray and the remaining black. He wonders: which leg did the dogs bite me, on the right or on the left?

Autolimio

^{ i }was born in 223, and died after living 84 years. Half of his life was spent in travel, a third was spent in pleasure. How much is a pound of nails worth and was Autolimius married?In the New Year, 200 men were taken out of the masquerade at the Bolshoi Theater per fight. If there were two hundred who fought, then how many were the abusive, the drunk, the slightly drunk and those who wanted to, but could not find the opportunity to fight?

What do you get after adding these figures?

20 boxes of tea were bought. In each box there were 5 puds

^{ ii }each pud had 40 pounds. Of the horses carrying the tea, two fell on the road, one of the coachmen became ill and 18 pounds were shed. The pound has 96 zolotniks^{ iii }of tea. He wonders, what difference is there between cucumber in brine and perplexity?The English language has 137,856,738 words, the French 0.7 more. The English joined the French and united both languages in one. He wonders, what is the third parrot worth and how long did it take to subjugate those peoples?

On Wednesday June 17, 1881, at 3 in the morning, the train must have left station A by the railroad , to arrive at station B at 11 at night but, before the same train departure, the order was received for the train to arrive at station В at 7 at night. Who loves the longest, the man or the woman?

My mother-in-law is 75 years old and my wife 42. What time is it?

Antosha Chejonté reported

If anyone knows the answer to any of these questions, would you be so kind as to share it? Thank you.

Let's move on to some more serious math. Two years later Chekhov published * The Repeater *. We are in Russia, at the end of the 19th century. Egor Ziberov, a seventh-grade student, gives private lessons to Petya, a 12-year-old boy. The boy's father attends the class; Egor wants to show off to ask for a raise. After Latin, comes the mathematics class (see [1]):

– Now let's move on to arithmetic. Take the board. What problem does it touch?

Petia spits on the blackboard, and then wipes it with her sleeve. The teacher takes the problem book and dictates:

–

A merchant bought one hundred thirty-eight sticks of black and blue cloth for five hundred and forty rubles. He wonders: How many sticks did he buy of one and the other if blue cost five rubles a stick and black three?You repeat the problem!Petia repeats it; and immediately begins, without saying a word, dividing five hundred and forty by one hundred and thirty-eight.

– Why do you divide? Wait!… Well, yes… go on. Is there residue? There doesn't have to be. Bring it on, I'll split it up. Ziberov divides it and gets three with remainder, which he is quick to erase.

«It's strange – thinks, pulling his hair and turning red – . How is this problem? … Hum! … It must be an indeterminate equation and not arithmetic – the teacher consults the solutions and finds seventy-five and sixty-three – . Hum! How strange! … Maybe you have to add five and three and then divide five hundred and forty by eight. Could it be that? … No, it's not that either.

– Good!… Please solve it once and for all! – says Petia.

– But why do you think about it so much? The problem is very simple! – says Udodov to Petya – What a fool you are, little boy! Solve it for yourself this time, Egor Alekseich!

Egor Alekseich takes the slate and prepares to solve it. Stutters It turns red. Pales.

– The point is that this problem is algebra – says – . It could be solved with X and with Y. And also without this … See? … I divide here … Do you understand? … Now this must be subtracted … Do you understand? …, or if not … It would be best if you bring it to me resolved tomorrow … Think about it!

Petia smiles maliciously; Udodov smiles too. They both understand the teacher's embarrassment. The seventh grader gets even more embarrassed, gets up and starts pacing the room.

– It can be solved without going to algebra – says Udodov. And reaching out to the

schioti^{ iv }he adds with a sigh – : Look. It is like this …Having made the calculations on the

schiotihe obtains the specified seventy-five and sixty-three without difficulty.– This is done our way … not scientific.

The teacher feels terribly uncomfortable. With his heart not beating, he looks at his watch and checks that the end of the lesson is still an hour and a quarter away. An eternity!

Indeed, the problem is simple. If X and Y designate, respectively, the number of rods of black cloth and that of blue, it is sufficient to propose a system of two equations with two unknowns ( X + Y = 138 and 3X + 5Y = 540), resulting in X = 75 and Y = 63. But Petia should not do it this way, she has not yet learned to solve systems of equations. Thus, you must find the solution using elementary methods. You can do this by assuming that if all the cloth had been blue, the 138 rods of cloth would have cost 5 × 138 = 690 rubles, that is 150 rubles (690-540 = 150) more than the actual amount. It is enough to note that the price difference between a blue cloth stick and a black stick is 2 rubles. Thus, dividing 150 by 2, we deduce that there must be 75 rods of black cloth and therefore the rest, 63 (138-75 = 63), must be blue. Simple arithmetic.

Even a “crazy mathematician” is sometimes unable to solve a simple problem without resorting to artifice. Udodov, with his abacus, finds the solution to the problem "in his own way …, unscientific" in the face of the embarrassed Egor who, probably, will be left without his pay raise …

** References **

[1] Anton Chekhov, * Complete stories (1880-1885) *Edition of Paul Viejo, Titivillus, 2013

[2] Yakov Perelman, * The Ancient Abacus and its Descendants * in Arithmetic recreational, 1954

Notes:

i ^{} Autolimio is a name invented by * Chekhov. *

ii ^{} The pud is a unit of mass equivalent to 40 Russian pounds, about 16.38 kilograms .

iii ^{} The zolotnik is an old Russian unit of mass equivalent to about 4.26 grams.

iv ^{} Russian abacus.

About the author: * Marta Macho Stadler is a professor of Topology in the Department of Mathematics of the UPV / EHU, and a regular contributor to ZTFNews, the blog of the Faculty of Science and Technology ogy of this university. *