Strange math question; Find $2 \mathbin\triangle 4 =\text{?}$

Question

$\newcommand{\d}{\mathbin{\triangle}}$How do you solve this question; $$1 \mathrel\triangle 0 = 3$$ $$1 \d 2 = 12$$ $$0 \d 2 = 5$$ $$1 \d 5 = 2 \d 3 = 96$$ Find $$2 \d 4 = \text{?}$$ Here it is the answer to the equation(please keep in mind that i need the method);

320

Answer 1

Let us write $f(x,y)=x \mathbin\triangle y$. Then by interpolation there is a polynomial $f(x,y)$ satisfying the conditions $$f(1,0)=3$$ $$f(1,2)=12$$ $$f(0,2)=5$$ $$f(1,5)=96$$ $$f(2,3)=96$$ for example

$$ f(x,y)=\frac{1}{110}(708x^3 + 621x^2y - 295xy^2 + 116y^3 - 378). $$ Then we have $$f(2,4)=\frac{6603}{55}.$$ This is not the only possibility, of course.

Answer 2

Questions like this are really exercises in guessing what the asker was thinking. It is not possible to determine the answer mathematically. Instead, you have to come up with some definition of $a \mathbin\triangle b$ that agrees with the information given. There are many such definitions. Here is one:

Let us represent the pair $(a,b)$ by $2^a 3^b$. Using the points $(2^1 3^0,3)$, $(2^1 3^2,12)$, $(2^0 3^2,5)$, $(2^1 3^5,96)$ and $(2^2 3^3, 96)$, we find the unique interpolating polynomial of minimal degree: $$ f(x) = \frac{150671}{272276484480}x^4 - \frac{19120391}{54455296896}x^3 + \frac{15349819}{378161784}x^2 - \frac{7010527}{56023968}x + \frac{72147351}{23343320}.$$ Now define $a \mathbin\triangle b = f(2^a3^b)$. We obtain $2 \mathbin\triangle 4 = -540445677/333476$.

Note that the polynomial $f$, called the Lagrange polynomial, can be found algorithmically, and so we have a general technique to automatically produce answers to questions like these.

Answer 3

Essentially, the task is to fill in the question mark in the following $\triangle$ operation table.

\begin{align} \triangle \quad&\quad 0&\quad 1&\quad\quad\quad2&\quad3&\quad\quad\quad4&\quad5\\ \hline\\ 0\quad&\quad*&*&\quad\quad\quad5&\quad*&\quad\quad\quad*&\quad*\\ 1\quad&\quad3&*&\quad\quad\quad12&\quad*&\quad\quad\quad*&\quad96\\ 2\quad&\quad*&*&\quad\quad\quad*&\quad96&\quad\quad\quad?&\quad*\\ \end{align}

Without further information about the $\triangle$ operation, one may pretty much fill in the missing spaces with random entries.

For example, one might add information such as:

"The operation $m\triangle n$ acts on non-negative integers $m$ and $n$ using a combination of the four operations of arithmetic plus exponentiation."

But even with that addendum, there might not be a unique answer.

Answer 4

A possible, somewhat simple, partial formulation is $a\triangle b = 2(a\triangle (b-1))$. This formulation is consistent with the three cases of $1\triangle b$. Using this formulation would give $2\triangle 4=2(2\triangle 3)=192$