Clock and Calendar Numbers

Transcript

Clock and Calendar Numbers This section explores interesting numerical patterns found in everyday time-keeping devices. It's divided into three parts: patterns in clock times, patterns in dates, and the repeating nature of calendars. 1. Clock Time Patterns The text points out that times on a 12-hour clock can form different types of patterns, giving examples like 4:44, 10:10, and 12:21. The puzzle is to find all possible times for each type. Type 1: Repeating Digits (e.g., 4:44) This is when all digits in the time are the same. Possible Times: 1:11, 2:22, 3:33, 4:44, 5:55. Type 2: Repeating Hour/Minute Sequence (e.g., 10:10) This is when the number for the hour matches the number for the minutes. Possible Times: 1:01, 2:02, 3:03, 4:04, 5:05, 6:06, 7:07, 8:08, 9:09, 10:10, 11:11, 12:12. Type 3: Palindromic Times (e.g., 12:21) This is when the time reads the same forwards and backwards (e.g., 12:21 -> 1221). Possible Times with 1-digit hours: 1:01, 1:11, 1:21, 1:31, 1:41, 1:51, and similarly for 2:_2, 3:_3, up to 9:_9. Possible Times with 2-digit hours: 10:01, 11:11, 12:21. 2. Date Patterns The section highlights two kinds of date patterns using birthdays as examples. Repeating Sequence Dates (Manish's Birthday: 20/12/2012) The pattern here is that the digits of the day and month (2012) are repeated to form the year (2012). The question asks for other such dates from the past. Examples: November 10, 1011 (10/11/1011) December 11, 1112 (11/12/1112) January 19, 1901 (19/01/1901) Palindromic Dates (Meghana's Birthday: 11/02/2011) This is when the full date written as a string of digits reads the same forwards and backwards (11022011). The question asks for all possible dates of this form from the past. These are rare. To form one, the year has to be a "reverse" of the month and day. Examples from the past: October 1, 1010 (01/10/1010) November 1, 1011 (01/11/1011) The last palindromic date before Meghana's was October 2, 2001 (02/10/2001). Let's check 02102001 - it is a palindrome. The next one after the example is February 2, 2020 (02/02/2020). 3. Reusing Calendars This part of the text asks if a calendar for one year can be reused for a future year. The answer is Yes! Why Calendars Change: A normal year has 365 days. Since 365 divided by 7 leaves a remainder of 1 (365 = 52 weeks + 1 day), a year usually starts one day of the week later than the previous year. Leap years have 366 days (52 weeks + 2 days), which shifts the start day by two days. When Calendars Repeat: For a calendar to be reused, a future year must start on the same day of the week and have the same leap year status (either both are leap years or both are not). Due to the pattern of these shifts, calendars repeat in cycles. Common cycles are 6 years, 11 years, and 28 years. For example, the calendar for 2023 (a non-leap year that started on a Sunday) can be reused in 2034. The calendar for 2024 (a leap year that started on a Monday) can be reused in 2052 (a 28-year cycle).