Number Grid Puzzle

Here is the recommended order to answer the questions from first to last: 1-Across , 1-Down , 2-Across , 2-Down .

Across

1. $4! = 24$ is the only factorial number that contains two-digit. Some terms in the factorial sequence are $1,2,6,24,120,720,\cdots$ . It's easy to see that factorial $n!$ increases as the integer $n$ increases.
2. Since the answer for 1-Down is $22$ (see 1-Down ), this forces $25$ to be the perfect square.

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Down

1. Because of the answer for 1-Across, the second-digit must be the 2. Thus, we have 22 as the palindrome .
2. If you have already answered all Across numbers correctly, the answer for 2-Down must be $45$ , which is the palindrome number. Otherwise, if you start with this problem, there is no easy way to determine the answer since there are in fact more than two-digit triangular number, such as $10$ and $55$ .

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Answer

The answer is $2 + 5 + 2 + 4 = \boxed{13}$ .