Segmented Time Cynthia Shepherd, Joanna Murakami, Jennifer Wright, Patrick Gass

1. hours minutes seconds 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 9:02’15’’ 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 questionWhat 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 questionHow 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 10:08.08 Minimum # of ticks is 10 at 1:11.11 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 8:08.08 and 12:08.08. 10 o’clock 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. 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/