Our sheik from last week, due to a mistake or perhaps to test the cunning of his children, did not leave them the entire herd, since 1/2 + 1/3 + 1/9 = 17/18, and that explains that the mullah recover his camel after giving it up so that the distribution would be possible without cutting up any animal.

As for the definition of an equation, for the first time (but I guess it won’t be the last) I’m going to give the answer of a robot, because our regular commenter pi science asked ChatGPT for it and got the following:

– An equation is a mathematical expression that establishes equality between two quantities or expressions.

– Definition in one word: equality.

– A physical simile to visualize an equation could be the scale for weighing objects. If you put the same weight on both sides of the scale, you would be establishing an equality between the weights. In the same way, in an equation, the expressions on both sides of the equals sign have the same value.

Arthur C. Clarke would cringe (he used to say that if we invented intelligent machines, it would be the last thing they would ever let us invent). In another order of things, an interesting problem circulated in the comments section last week, not suitable for anarithmetics, of which I offer a slightly simpler variant:

I have a certain number of consecutive bingo balls, starting with 1, and I see that I can group them into two groups of consecutive balls that have the same sum. How many balls do I have? (The trivial solution 3 is not valid, because although balls 1 and 2 add up to the same as 3, there are not two groups of balls, in the plural).

### Pell’s equation

“Say, mathematician, what is the square that multiplied by 8 becomes, together with the unit, another square?”

If you find it, you will have solved a famous problem posed by the Indian mathematician Bhaskara in the seventh century. A problem that, with the current notation, we would express as follows: 8y² + 1 = x² , that is, x² – 8y² = 1, which is a Pell equation (named after the 17th century English mathematician John Pell), whose general form is x² – ny²= 1.

A thousand years before Bhaskara, Archimedes had already studied this type of equations in relation to the so-called “cattle problem” or “Sun herd problem”, which reads as follows:

In the herd of the Sun god there are white, black, spotted and brown bulls, and cows of the same four types.

The number of white bulls is half and one third of the blacks plus the browns.

The number of black bulls is equal to a quarter plus a fifth of the spotted ones plus the brown ones.

The number of spotted bulls is equal to one sixth plus one seventh of the whites plus the browns.

The number of white cows is equal to one third plus one fourth of the sum of the black bulls and the black cows.

The number of black cows is equal to one quarter plus one fifth of the sum of the spotted bulls plus the spotted cows.

The number of spotted cows is equal to one fifth plus one sixth of the sum of the brown bulls plus the brown cows.

The number of brown cows is equal to the sixth plus the seventh part of the sum of the white bulls plus the white cows.

The sum of the black and white tori is a square number (ie a perfect square).

The sum of the speckled and brown tori is a triangular number (ie, of the form n(n+1)/2).

It seems that the problem was raised by Archimedes to his friend Eratosthenes, and it is not easy. Don’t try to figure it out by trial and error, for, as befits a god’s herd, there are several million cattle of each type.

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