While multiplying three real numbers, Ashok took one of the numbers as $73$ instead of $37$. As a result, the product went up by $720$. Then the minimum possible value of the sum of squares of the other two numbers is

Let us call the other two numbers $y$ and $z$.

The three numbers were supposed to be $37$, $y$, and $z$ with original product $37yz$.

Ashok used $73$ instead of $37$, giving incorrect product $73yz$.

The product increased by $720$, so:

$73yz - 37yz = 720$

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$73yz - 37yz = (73-37)yz = 36yz = 720$

$yz = \frac{720}{36} = 20$

The product of the other two numbers must equal $20$.

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We need to find the minimum value of $y^2 + z^2$ given that $yz = 20$.

For any two numbers with a fixed product, their sum of squares is minimized when the numbers are equal.

Since $yz = 20$, we can write $z = \frac{20}{y}$.

So $y^2 + z^2 = y^2 + \left(\frac{20}{y}\right)^2 = y^2 + \frac{400}{y^2}$

The minimum occurs when $y^2 = \frac{400}{y^2}$, which gives us $y^4 = 400$, so $y^2 = 20$.

When $y = z$, we have $y \times y = y^2 = 20$, so $y^2 = 20$ and $z^2 = 20$.

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When $y = z$ and $yz = 20$:

$y^2 = 20$ and $z^2 = 20$

Therefore: $y^2 + z^2 = 20 + 20 = 40$

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The minimum possible value of the sum of squares of the other two numbers is $40$.