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A math problem is able to have multiple right solutions

This "riddle" has raised more than one million comments on Facebook.

What would be your answer to the question mark? The most common answers are 40 and 96... Does anyone could say why? Hence if it is given the first terms in a sequence and the sequence is not well defined, the next term is able has multiple right solutions, even infinite solutions. For instance, in the next problem Find the solution which I gave one solution, is able to have infinite answers, even the other answers given in the problem can be right... Just only have to take polynomials with two variables x,y which satisfies the conditions, for example,can you find a polynomial with variables x, y satisfying the first conditions and such that f(13,4) = 123 or f(10,9) = 123?... So my conclusion is this one: Can you post one riddle where different right solutions are valid?and why my "riddle" is able to have 40 or 96 as valid solutions?can you find other right solution and say why?

Note by Guillermo Templado
5 months ago

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I think they expect 96 as the answer if they do this. \(a+b=a×(b+1)\) But like @Nihar Mahajan said you can have \(\infty\) solutions Vignesh S · 5 months ago

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The riddle has infinitely many solutions using LZOB or Lagrange interpolation, since the choices are not provided. Nihar Mahajan · 5 months ago

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@Nihar Mahajan Exactly, for the problem Find the solution you can use Lagrange interpolation with two variables, please wait a bit, and I'll give the answer Guillermo Templado · 5 months ago

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@Nihar Mahajan please, be calm. I can not answer to everybody Nihar Mahajan · 5 months ago

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@Nihar Mahajan We have \( f(5,3) = 28, \space f(12,10) = 222 \space , f(9,1) = 810, \space f(4,2) = 26, \space f(8,3) = 511, f(10,1) = 911, \space f(7,3) = 410\). This problem using Lagrange interpolation is able to have \(\infty\) answers. For example, I want furthemore \(f(13,9) = 123.\) Is evrything allright so far? haha. Ok, I'm going to look for a polynomial with variables x,y and some constants a,b,c,d,e,f,g,h,i, ...fullfiling these conditions,(notice I'm going to repeat this factor (y -1) and (y - 3), I would not to do it, but I'm going to do it for clarity) so the polynomial has to be \(\small f(x,y) = a(x -5)(y -3)(x -12)(y -10)(x - 9)(y -1)(x - 4)(y -2)(x - 8)(y -3)(x - 10)(y -1)(x - 7)(y - 3) + \) \(\small + b(x -5)(y -3)(x -12)(y -10)(x - 9)(y -1)(x - 4)(y -2)(x - 8)(y -3)(x - 10)(y -1)(x - 13)(y -9) + \) \(\small + c(x -5)(y -3)(x -12)(y -10)(x - 9)(y -1)(x - 4)(y -2)(x - 8)(y -3)(x - 7)(y - 3)(x -13)(y - 9) +\) \(\small + d(x -5)(y -3)(x -12)(y -10)(x - 9)(y -1)(x - 4)(y -2)(x - 10)(y -1)(x - 7)(y - 3)(x - 13)(y - 9) +... \) and get the constants a,b ,c ,d e,f, g, h,... fulfilling the requisites... so this problem has \(\infty\) right solutions... Ok, now I'm going to eat... Later I'll review the comments Guillermo Templado · 5 months ago

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@Nihar Mahajan How is the answer 96 or 40? Abhay Kumar · 5 months ago

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@Abhay Kumar I got 96 but not 40. Nihar Mahajan · 5 months ago

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@Nihar Mahajan ok, 96 can be got of this form, \(\begin{cases}1 + 4 = 1 \cdot 4 + 1 = 5 \\ 2 + 5 = 2 \cdot 5 + 2 = 12 \\ 3 + 6 = 3 \cdot 6 + 3 = 21\end{cases}\) so I can say \(8 + 11 = 8 \cdot 11 + 8 = 96\). Now, we are going to get 40,ok...please, be cal, wait my answer,haha.. Guillermo Templado · 5 months ago

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@Guillermo Templado Wait, I will wait. (LOL) Nihar Mahajan · 5 months ago

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@Nihar Mahajan Now, we are going 40.. We can consider (1,4), (2,5), (3,6) and (8,11) to be the first terms of one sequence in \(\mathbb{N}^2\) and \(1 + 4 = 5\) and we can define for the next terms its sum added to the sum obtained before. Hence \(2 + 5 = 7 + \text{ sum obtained before } = 7 + 5 = 12\) , now \( 3 + 6 = 9 + \text{ sum obtained before } = 9 + 12 = 21\) so \(8 + 11 = 19 + \text{ sum obtained before } = 19 + 21 = 40 \) Guillermo Templado · 5 months ago

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@Guillermo Templado Oh! I considered \(8,11\) as the eighth term! Nihar Mahajan · 5 months ago

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@Nihar Mahajan but, they can be considered the first terms in a sequence. Now I'm going with the Lagrange interpolation Guillermo Templado · 5 months ago

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@Abhay Kumar please, be calm. I can not answer to everybody Guillermo Templado · 5 months ago

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@Nihar Mahajan Why are you assuming the function is a polynomial? (just asking) Deeparaj Bhat · 5 months ago

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@Deeparaj Bhat Its not wrong to assume a polynomial right? Vignesh S · 5 months ago

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@Vignesh S I could be any arbitrary function of two variables for that matter as we've not been given that the function IS a polynomial. Deeparaj Bhat · 5 months ago

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@Deeparaj Bhat please, be calm. I can not answer to everybody Guillermo Templado · 5 months ago

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@Deeparaj Bhat So it can be \(\infty\) if its a polynomial and if its some other function it can have finite or infinite. Therefore in general it can be said infinite Vignesh S · 5 months ago

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@Vignesh S please, be calm. I can not answer to everybody.... Guillermo Templado · 5 months ago

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@Guillermo Templado Hey I was asking @Deeparaj Bhat. And it was a comment and not a question Vignesh S · 5 months ago

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@Vignesh S ok, sorry, my apologies Guillermo Templado · 5 months ago

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The answer is 40. :P
In this riddle the answer of the previous statement is added to the terms of the original statement.
First statement starts with 1 + 4 = \(\color{red}{\text{5}}\).
Second statement:- \(\color{red}{\text{5}}\) + 2 + 5 = \(\color {blue}{\text {12}}\).
Third statement:- \(\color{blue}{\text {12}}\) + 3 + 6 = \(\color{green}{\text {21}}\).
Fourth statement should be \(\color{green}{\text {21}}\) + 8 + 11 = \(\boxed{40}\). Ashish Siva · 5 months ago

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