# Problem 9.4 (7 points)

A palindrome is a number, letter combination, word, or text that reads the same in both directions. How much time in a day do palindromes appear on the clock display, if the clock shows time from 00.00.00 to 23.59.59?
# Solution:

If the digits on the display are ab.cd.mn, then $\mathrm{a}=0,1,2,0 \leq \mathrm{b} \leq 9,0 \leq \mathrm{c} \leq 5,0 \leq \mathrm{d} \leq 9,0 \leq \mathrm{m} \leq 5$, $0 \leq \mathrm{n} \leq 9$. Therefore, if $\mathrm{a}=\mathrm{n}, \mathrm{b}=\mathrm{m}, \mathrm{c}=\mathrm{d}$, then the symmetric number on the display is uniquely determined by the digits $\mathrm{a}, \mathrm{b}$, and $\mathrm{c}$, where $\mathrm{a}=0,1,2 ; 0 \leq \mathrm{b} \leq 5 ; 0 \leq \mathrm{c} \leq 5$. If $\mathrm{a}=0$ or 1, then $\mathrm{b}$ and $\mathrm{c}$ can be any digits from 0 to 5, the number of such sets of numbers is $2 \cdot 6 \cdot 6=72$. If $\mathrm{a}=2$, then $\mathrm{b}=0,1,2,3 ; 0 \leq \mathrm{c} \leq 5$, the number of such sets is $4 \cdot 6=24$. In total, $72+24=96$ sets of numbers, each of which is displayed for 1 second.

| Criteria | Points |
| :--- | :---: |
| Complete solution | 7 |
| Correct approach with arithmetic error | 5 |
| Correctly specified numerical ranges for $\mathrm{a}, \mathrm{b}, \mathrm{c}$. | 3 |
| Correct answer without justification | 1 |

## Answer: 96 seconds