Take Your Time!

Hour $0$ , only $1$ moment ( $1^2$ ): $00:00:00$ ,

Hour $1, 4$ moments ( $2^2$ ): $01:01:01, 01:01:10, 01:10:01, 01:10:10$ ,

Hour $2, 9$ moments ( $3^2$ ): $02:02:02, 02:02:11, 02:02:20, \text{ etc } \ldots$ ,

Hour $3, 16$ moments ( $4^2$ ): $\ldots$ ,

There is clearly a pattern: the squares! However , there can not be anything above $36$ moments for any given hour because we are dealing with time! Example:

Hour $17, 36$ moments:

$17:(08 \text{ or } 17 \text{ or } 26 \text{ or } 35 \text{ or } 44 \text{ or } 53) : (08 \text{ or } 17 \text{ or } 26 \text{ or } 35 \text{ or } 44 \text{ or } 53)$ .

There is also the special case of $19$ to consider— $\text{digsum}(19)=10$ , and the only possible combinations for this are $19, 28, 37, 46, 54$ , so there are only $25$ moments instead of $36$ .

Hence, we have

$1+4+9+16+25+36+36+36+36+36+4+9+16+25+36+$ $36+36+36+36+25+9+16+25+36=A=580$ , and $\lceil 10^{-2} \cdot 580 \rceil=\large{\boxed{6}}$ .