Abstract: A digital LCD watch displays hours, minutes, and seconds in AM/PM mode. Each LCD number displayed has a certain number of segments turned on. For example, the number 1 has two segments, and at 9:02

1
Segmented Time Cynthia Shepherd, Joanna
Murakami, Jennifer Wright, Patrick Gass Prof.
Fernandez and Prof. Abrego (Advisors) California
State University Northridge Math Club Officers
Expected Ticks
Abstract A digital LCD watch
displays hours, minutes, and seconds in AM/PM
mode. Each LCD number displayed has a certain
number of segments turned on. For example, the
number 1 has two segments, and at 90215 there
are 24 segments turned on. How many times during
the day are there exactly 33 segments turned on?
What about another number of segments? It would
also be useful to find the total amount of energy
spent during one day. This is equivalent to
finding the expected number of segments turned on
during a day.
24 Hour Clock
New question What about some other number of
ticks?
Lets apply this program to a 24 hour clock
In order to make the same calculations for a
24-hour clock, we need only make a few changes to
the program. Only the range of the hours is
different, so the possible number of ticks in the
hours component range is from 2 to 11.
We know that Max 34 ticks happens 2 times a
day 33 ticks happens 28 times a day Min 10 ticks
happens 2 times a day. There are 24x60x60 86400
different times in a day. It would take quite a
bit of time to find the distribution of ticks, by
hand. We need a better way.
Counting The Ticks
Initial question How many times during the day
are there exactly 33 segments (ticks) turned on?
Maximum of ticks to be turned on at one time is
34 at 1008.08 Minimum of ticks is 10 at
111.11
HOURS HOURS MINUTES MINUTES SECONDS SECONDS SECONDS
tens ones tens ones
___ ___ ___ ___ . ___ ___

Time Tick Time Tick Time Tick Time Tick Time Tick
1 2 0 6 0 6 0 6 0 6
2 5 1 2 1 2 1 2 1 2
3 5 2 5 2 5 2 5 2 5
4 4 3 5 3 5 3 5 3 5
5 5 4 4 4 4 4 4 4 4
6 6 5 5 5 5 5 5 5 5
7 3 6 6 6 6
8 7 7 3 7 3
9 6 8 7 8 7
10 8 9 6 9 6
11 4
12 7
Also, we have to change the vector H, which
contains the probabilities of each number of
ticks in the hours slot.
We then compared the distribution of ticks in a
12-hour clock to the distribution of ticks in a
24-hour clock.
12-hour clock in black 24-hour clock in red
Distribution of Ticks
Maximum ticks for seconds is 13. The same is true
for minutes.
To find the distribution of ticks, we will graph
n (ticks) versus the probability of n, where n is
the number of ticks ranging from the min of 10 to
the max of 34. The probability of n is
If we maximize minutes and seconds, we will have
26 ticks. To reach 33 ticks we will need at
least 7 tick for the hour. The only that fill
this requirement are the hours of 8, 10, and 12.
Since 8 and 12 are exactly 7 ticks, the times
are 808.08 and 1208.08. 10 oclock has 8
ticks so we need one less tick in the minutes or
seconds. So we first take a tick from the
minutes and them one from the seconds.
Using the poss24 function along with a summing
function, we were able to obtain the expected
number of ticks in a day for a 24-hour clock.
List of possible times with 33 ticks.
Due to the extensive calculations required to
compute this for each n, we wrote a program in
Mathematica to simplify the process.
808.08 1008.09 1000.08 1038.08
1208.08 1008.28 1006.08 1058.08
1008.00 1008.38 1009.08
1008.06 1008.58 1028.08
Conclusion It is easy to find the power used
now that we know the expectation of ticks. The
12-hour clock will use less energy than a 24-hour
clock. It is also possible to modify the
Mathematica code to calculate other types of LCD
clocks that include the date or day of the week.
Fig. Distribution of ticks.
In a 24 hours period there is exactly 33 ticks
28 times a day.
This problem can be found at http//www.csun.edu/
math/probweek/spring05/projects/projects09s05.pdf
Similar problems can be found at
http//www.csun.edu/math/probweek/