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$$\begin{array}{|c|c|c|}\hline 2&11&2243\\\hline4&19&3739\\\hline 5&23&4487\\\hline 3&186&?\\\hline\end{array}$$

What is the missing number ?

I have searched for days but still no results, the only thing I noticed is this: $$4-2=2=2.1=2.(5-4)$$ $$19-11=8=2.4=2.(23-19)$$ $$3739-2243=1496=2.748=2.(4487-3739)$$ But it has nothing to do with the numbers at the end. I think it is the value of some polynomial $f(x)$ with many values of $x$. For example: $f(x)=\dfrac{1}{12}x^4-\dfrac{5}{6}n^3+\dfrac{47}{12}n^2-\dfrac{25}{6}n+2$, with the values of $x$ being $1, 2, 3, 4, 5$ respectively,$f(x)$ has values $1, 4, 9, 16, 27$ respectively

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    If you accept a polynomial, you can make a polynomial have any group of ordered pairs you want. https://en.wikipedia.org/wiki/Lagrange_polynomial Thus you could make the last number anything, using that logic. – Alan Jul 15 '21 at 03:13
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    One small thing I notice: If you double the values in the first row and add one, you get the values in the third row. – Semiclassical Jul 15 '21 at 03:18
  • Taking it by rows: 2x11 = 22, 2x2x11-1 = 43. Second row:2x19=38, and you get 38-1 followed by 38+1. Third row is a bit more off, double 23 is 46, but double the 44 is 88 and subtract 1. It's rough, not exact. – RobertTheTutor Jul 15 '21 at 03:18
  • Following up on my prior comment, in the first three rows we have the following: Denoting the first value as $x$, the second value is $4(x+1)-1$ and the third is $748(x+1)-1$. (Equivalently: if you add 1 to the first three rows, then the second and third columns are 4 and 748 times the first column respectively.) But this falls apart on the last row: the second value would need to be 15 to continue the pattern. – Semiclassical Jul 15 '21 at 03:33
  • I see that if the last number in the column is subtracted from each of the upper three numbers in the next column to the right, they are then each divisible by the number to their left, with equal quotients. – abiessu Jul 15 '21 at 05:15
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    Or, multiply the top two numbers and subtract the third. – abiessu Jul 15 '21 at 05:16

1 Answers1

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A simple rule that appears to have meaning: multiply the top two numbers in a column and subtract the third to get the fourth number. Final number: $8382090$.

abiessu
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