The Guide's Question
In January 1988, I escaped the freezing cold of New York City for a week by taking a seven-day Caribbean cruise. One of the stops on the cruise was Cancun, Mexico.
At that port, I elected to take an optional side trip to the Maya pyramids at Chichen Itza. The trip to that famous archeological site was in an air-conditioned bus with free Coca-Cola (and other soft drinks) and a guide.
The guide explained that he was pure Maya, with no Spanish ancestors in his family. He seemed quite proud of that. He said that he was going to explain something about the Maya calendar system which, he said, was more accurate than the Gregorian Calendar now in use worldwide. In particular, the Gregorian Calendar uses 365 97/400 for the number of days in a solar year and the Maya even in the year 1000 had a better, more exact estimate than that.
In spite of having that better estimate, the Maya used a secular calendar of exactly 365 days. The guide said that they liked the idea that the equinoxes and solstices would process through the secular year in a cycle that would last over 1400 years. He explained that they loved long cycles and extended their calendar over a million years into the future.
The guide explained that they also had a religious calendar which was kept separately from and simultaneously with their secular calendar. He explained that they had 20 gods and that their religious calendar had 13 months with 20 days in each month, each day being named for one of their gods, thus making 260 days in their religious calendar.
The guide then said it was a myth that the Maya executed people on their 52nd secular birthday, but they did believe that 52 secular years was the gods' allotted lifespan (the same way that 70, three score and ten years, is the Christian bible's allotted lifespan).
The guide then paused and asked the question: Why did the Maya believe that 52 of their secular years was the allotted lifespan? Can anybody tell me why?
After about ten seconds, I raised my hand.
"Yes," said the guide. "You have an answer?"
"Well," I said, "you have told us that the Maya used two calendars, one secular with 365 days and the other religious with 260 days. Now, 365 = 5x73 and 260 = 5x52. 73 and 52 have no common factor above 1. Therefore, the least common multiple of 365 and 260 is 52x365 = 73x260. This means that anybody's 52nd secular birthday will also be their 73rd religious birthday, and that is the first time that they will have such a double birthday."
The guide's jaw dropped. He looked stunned. He then said, "I have been giving this tour for ten years now, six days a week. I have always asked this same question about the allotted lifespan. Today is the first time that anyone gave a complete, correct answer. Please tell me, sir, where did you get your Ph.D. in mathematics?"
I could have been nasty and said something like, "I knew about least common multiples and greatest common divisors when I was 12 years old. I could have answered your question at age 12, before taking high school algebra."
But that would have made everybody on the bus feel ashamed that they hadn't figured it out. So, I actually just answered his question and said, "I got my Ph.D. in mathematics in May 1976 at M.I.T."
Some people on the bus gasped and some actually applauded. The guide came and sat next to me. We talked about the mathematics known by the Maya one thousand years ago for the rest of the trip to Chichen Itza.
In January 1988, I escaped the freezing cold of New York City for a week by taking a seven-day Caribbean cruise. One of the stops on the cruise was Cancun, Mexico.
At that port, I elected to take an optional side trip to the Maya pyramids at Chichen Itza. The trip to that famous archeological site was in an air-conditioned bus with free Coca-Cola (and other soft drinks) and a guide.
The guide explained that he was pure Maya, with no Spanish ancestors in his family. He seemed quite proud of that. He said that he was going to explain something about the Maya calendar system which, he said, was more accurate than the Gregorian Calendar now in use worldwide. In particular, the Gregorian Calendar uses 365 97/400 for the number of days in a solar year and the Maya even in the year 1000 had a better, more exact estimate than that.
In spite of having that better estimate, the Maya used a secular calendar of exactly 365 days. The guide said that they liked the idea that the equinoxes and solstices would process through the secular year in a cycle that would last over 1400 years. He explained that they loved long cycles and extended their calendar over a million years into the future.
The guide explained that they also had a religious calendar which was kept separately from and simultaneously with their secular calendar. He explained that they had 20 gods and that their religious calendar had 13 months with 20 days in each month, each day being named for one of their gods, thus making 260 days in their religious calendar.
The guide then said it was a myth that the Maya executed people on their 52nd secular birthday, but they did believe that 52 secular years was the gods' allotted lifespan (the same way that 70, three score and ten years, is the Christian bible's allotted lifespan).
The guide then paused and asked the question: Why did the Maya believe that 52 of their secular years was the allotted lifespan? Can anybody tell me why?
After about ten seconds, I raised my hand.
"Yes," said the guide. "You have an answer?"
"Well," I said, "you have told us that the Maya used two calendars, one secular with 365 days and the other religious with 260 days. Now, 365 = 5x73 and 260 = 5x52. 73 and 52 have no common factor above 1. Therefore, the least common multiple of 365 and 260 is 52x365 = 73x260. This means that anybody's 52nd secular birthday will also be their 73rd religious birthday, and that is the first time that they will have such a double birthday."
The guide's jaw dropped. He looked stunned. He then said, "I have been giving this tour for ten years now, six days a week. I have always asked this same question about the allotted lifespan. Today is the first time that anyone gave a complete, correct answer. Please tell me, sir, where did you get your Ph.D. in mathematics?"
I could have been nasty and said something like, "I knew about least common multiples and greatest common divisors when I was 12 years old. I could have answered your question at age 12, before taking high school algebra."
But that would have made everybody on the bus feel ashamed that they hadn't figured it out. So, I actually just answered his question and said, "I got my Ph.D. in mathematics in May 1976 at M.I.T."
Some people on the bus gasped and some actually applauded. The guide came and sat next to me. We talked about the mathematics known by the Maya one thousand years ago for the rest of the trip to Chichen Itza.
I hope that there is some pizza at the party which I...