# How often does a digital watch show a palindrome?

A palindrome is a word, phrase, number, or other sequence of characters which reads the same backward or forward, such as madam or kayak.

We have a digital watch showing the time HH:MM:SS as shown in the figure below:

How many times in a 24-hour period of time on a digital clock does the number reveal a palindrome?

For example: $01:11:10$

• why did you remove the PM? I had a thing going with it. :D – Marius Jan 10 '17 at 12:25
• Does the font matter? – Nautilus Jan 10 '17 at 13:34
• Does this watch support the leap second ? eg 23:59:60 :) – nl-x Jan 10 '17 at 13:41
• Never mind, apparently it doesn't. – Nautilus Jan 10 '17 at 13:47
• @nl-x, will there ever be a palindrome leap second? :P – Wildcard Jan 11 '17 at 5:18

Let us denote the time by AB:CD:EF, where AB is between 00 and 23, CD and EF are each between 00 and 59, and of course A=F, B=E, C=D.

The middle two digits

Since C=D and CD ranges from 00 to 59, there are six possibilities: 00, 11, 22, 33, 44, 55.

The outer four digits

The first two digits (AB) range from 00 to 23. How many of these possibilities give valid possibilities for the last two digits (EF) when reversed? We know EF ranges from 00 to 59, so B must be between 0 and 5. Thus the possibilities for AB are 00 to 05, 10 to 15, 20 to 23 - sixteen possibilities in total.

Six times sixteen is ninety-six.

I know this has a valid answer but I noticed something.
[EDIT] I noticed it before the OP sabotaged me and removed it. I will leave this answer here though because I'm really proud of my finding and it makes me feel smart :D. See original image here [/EDIT]

The clock in the image shows PM. This means that the clock will only show hours between 1 and 12, twice. Once for AM and once for PM. It will never show the hours 13 to 23 or 00.

This means that the combinations are:

For minutes, as @rand al'thor said 00, 11, 22, 33, 44, 55 but the hours that can be displayed and has a corespondence to the seconds in the reverse order are:
01, 02, 03, 04, 05, 10, 11 and 12. (8 combinations)

This means that the total number of combinations is:

minutes combination * hours combimation * 2 (2 sets of 12h).
$6 \times 8 \times 2 = 96$.

• good catch :) Fixed it. – Oray Jan 10 '17 at 12:24
• @Oray It depends on how you look at it. From my point of view you broke it. :D – Marius Jan 10 '17 at 12:27
• On most 12-hour displays i've seen, leading zeroes are not shown. 9:02:37 rather than 09:02:37. This means palindromes before 10am would be 5-digit ones like 9:14:19 . – IanF1 Jan 10 '17 at 19:35
• Most but not all :) – Marius Jan 10 '17 at 19:39
• Bizarre that it actually doesn't matter whether the watch is 12 hour or 24 hour, the answer is still the same. – Wildcard Jan 11 '17 at 5:20

The more obvious answer that doesn't take the font shape into account:

The time must be written like $AB:CC:BA$. Since $A$ is the first hour digit, it's less than 3. Being the first second digit, $B<6$ (same for $C$). If $A<2$, there are 2 values available for $A$, 6 for $B$ and $C$, so there are 72 combinations following this rule. If $A=2$, there are 4 values available for $B$ and 6 for $C$, which means 96 combinations in total.

Assuming the font plays a role:

The vertically symmetrical digits for each one:

0 - 0
1 - 1
2 - 5
8 - 8.

At $AB:CD:EF$, $A<3; C,E<6$. If $A<2$, there are 2 values available for $A$, with whichever chosen also used for $F$. $B$ and $C$ can be 0, 1, 2 or 5 ($2*4*4=32$ combinations). If $A=2$, then $B=0,1$ or $2$, and $C=0,1,2$ or $5$ ($3*4=12$ combinations). There are 44 in total.

Edit: the last one wasn't what the OP meant.

• What does vertical symmetry have to do with palindromicity? – Rand al'Thor Jan 10 '17 at 13:35

It's all down to the minutes - it needs a palindromic number of minutes.

00, 11, 22, 33, 44, 55 So for every hour, there are six possible palindromes.

Then work out which hours can have a reversed number of seconds (less than 60):

01, 02, 03, 04, 05 (06, 07, 08, 09) 10, 11, 12, 13, 14, 15 (16, 17, 18, 19) 20, 21, 22, 24, 24. 16 possibles. 16 x 6 = 96

In the am/pm scenario, the preceding zero would never be included, therefore the number of palindromes possible would be significantly reduced below 96. Also, since the AM and PM symbols cannot be part of the palindrome, the first half, and the second half, of the 24 hour period would be indistinguishable, and therefore not add to the total.

• Which adds a new set of possible five-digit palindromes to the mix. – ron Jan 10 '17 at 19:44
• Well, it does indeed add new and interesting five-digit possibilities...so where is your calculation of those possibilities? This isn't really an answer as currently written. (Stack Exchange isn't a discussion forum.) – Wildcard Jan 11 '17 at 5:22
• See the OP's question. 01:11:10 is included as an example, so obviously the preceding 0 is included. – boboquack Jan 11 '17 at 5:41