[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 427

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Well they had to figure out the number their opponent heard somehow.

A: "I can't identify the numbers".
B: "I knew in advance that you couldn't identify the numbers".

B's answer is just a provocative sneer. So the solution is very simple.

 

I think I've figured it out. Since we fit the cells into a square and each cell is a square, we have the right to divide the whole piece by 4 in sequence until we get the number of cells = A*A

The only difficulty will be the consumption of the material - the whole without remainder.

 

Let's reason. If Sage A is given the product of 22, can he guess both numbers? Yes: 2 and 11, because 22 is the only way to decompose into the product of numbers greater than 1.

How about 24? No.

I wonder how to decipher B's snide remark.

 
Divide a hundred metre piece by 4 - you get 4 pieces of 25 metres each. That's 1 square. Now divide each of these pieces by 4 again to make 4 squares. If you connect them to the grid, then 4 pieces will be superfluous - adjacent walls. That's the problem. This means that if we don't know the number of cells beforehand we cannot get the equation of the first degree. If we do, but we don't know the cell size, the problem is solvable.
 
Mathemat:

Let's reason. If sage A is given the product of 22, can he guess both numbers? Yes: 2 and 11, because 22 is the only way to decompose into the product of numbers greater than 1.

How about 24? No.

I wonder how to decipher B's snide remark.


If the product of the conceived numbers is not equal to the sum, then each wise man has only half of the problem at his disposal. He has to decompose the resulting number into its summands or factors. The result will be a system of two equations with three unknowns. The only thing left is either brute force (which you won't do quickly) or the problem will not have a mathematically rigorous solution. The system will consist, for example, of the following equations

a*b=22

a+b=c

 
And of course, the provocative remark of one of the opponents could only come out if one gets a simple solution - that is, one does not have to go through the multiplicators, painfully trying to solve a system of equations with three unknowns. Such a search will not come easily. Well, judging by the context of the situation, it took them very little time and effort to think it through.
 

Why so fast, ValS? Delete the post, let's think together :)

2 drknn: Once again: Sage A speaks first, after getting the number 22. He would immediately say he guessed it. So it's not numbers 2 and 11.

P.S. I haven't looked at ValS's post yet. What does phrase B mean? How does he know in advance that A won't guess the numbers when he gets the sum? This is a very capacious answer in fact, it contains almost all the information about the numbers! It means that any decomposition of the sum reported by B into two summands results in at least one of the summands containing two multipliers. Or so it goes.

 
Mathemat:
Why so fast, ValS? Delete the post, let's think together :)

The solution he gives relies on the fact that the conceived numbers are different. This reasoning is incomplete, because it does not take into account the possibility of conceiving two identical numbers, which does not contradict the condition of the problem.
 
Easy))
 
drknn:

The solution he gives relies on the fact that the conceived numbers are different. This reasoning is incomplete, because it does not take into account the possibility of conceiving two identical numbers, which does not contradict the condition of the problem.

Do you suppose that something will change?
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