You are about to read a special story problem, one I call a “mathematical fairy story” problem. You may find it a little hard at first, but that’s okay. If a story problem is easy for you to do, then it really isn’t a “problem” anymore, is it?”
Piet Hein, a Danish mathematician, architect, poet, philosopher, is famous for his little short poems, called Grooks, that contain little bits of wisdom. My favorite one says:
Problems Worthy of attack, Prove their worth By hitting back.
So if my problem below “hits back”, then it has proved its worth.
- 1. A + B is a square number.
2. (A – B) – G = 0.
Well, that’s basically where our story ends. Now it’s your turn to do some serious puzzling and problem solving, by posing some questions for you to ponder:
- Can you find a logical explanation to account for the change of expressions on Ms. Omega’s face as she read the solutions submitted by the three children?
- Regarding the naming of the students, we will tell you that the ratio of male to female individuals present in the classroom during this perplexing incident was 1:1. (I guess it’s only fair to warn you that the teacher is one of the individuals being counted.) If Papi Pieros, in his role as school principal, were to assign one more new student to this particular section, Ms. Omega would have a little trouble assigning a special name to him/her. But since that hasn’t happened yet, tell us the number of boys in this group.
- Also regarding the naming system applied by the teacher, another strange oddity is evident. Some of the students’ names become palindromes when written in the style described in the story. How many such cases of palindromic names are there? List them.
- The late Paul Erdös, considered by many to be the most prolific writer of articles about mathematics of this century, perhaps of all time, never married, nor had a family of his own. Yet he dearly loved young children and they loved him. He used one of the Greek letters as his affectionate way to refer to them. What was that letter and why did Erdös undoubtedly choose that particular letter?
- This story was written with a special theme in mind. It will be rather subtle for some readers to deduce, or very mysterious, if you will, for others. And maybe impossible for the remaining people to figure out at all. But there are, we believe, plenty of clues scattered about to provide the necessary information for the most intrepid of problem solvers. Which pigeonhole do you fit in?
- The title of this story even has a familiar ring to it, doesn’t it seem to you? Could you draw a connection between the two that makes delicious sense in the culinary arts?
If your wish to send in solutions to some or all of the questions posed above, we encourage it.
Here are my two main addresses: firstname.lastname@example.org or email@example.com.
The following day, after the pupils finished passing into the classroom and getting seated, who should enter but none other than Papi Parios himself! He often visited the classes in his schools, (partially to check up on his staff to see if they were promoting his particular philosophy – but he also enjoyed the experience of learning something new about math).
Paying no attention to his presence, Alpha and Beta raised their hands, requesting permission to speak. Ms. Omega turned to Alpha first, “Yes, Alpha, what’s on your mind today?”
“Well, Ms. O,” she said (using her teacher’s proper last name due to the fact that the principal was looking on), “after yesterday’s inspiring lesson, Beta and I have been playing around with our own little puzzle, using three primes. May we show it to you on the chalkboard?” “Of course, you may. Come forward, girls.”
Alpha took the chalk first, speaking as she wrote the numbers, “We have discovered something peculiar about the first three consecutive primes: 2, 3, and 5. If we multiply the larger pair, 3 x 5, we get 15. Now add 1 to obtain 16. Finally, we divide by the smallest prime, 2. And the answer, 8, is integral!’ (Alpha always loved to show off by using fancy, high level terminology.)
“Very nicely explained, Alpha,” said Ms. Omega. “But by the very basic principles of odd and even numbers, you can always expect that kind of result. Can you see why?”
“Yes,” Alpha replied. “We already thought of that. Here is our formula as proof.” Turning once again to the board, she wrote:
odd x odd + 1 odd + 1 even ----------------- = ------------ = -------- = integer (Q.E.D.) 2 2 2
“Now that’s impressive, indeed!” beamed Ms. Omega. From the back of the room, Papi Parios began applauding enthusiastically.
“But that’s not all we discovered, Ms. O,” interjected Beta, with obvious pride in her voice. “Our pattern works for the next set of three consecutive primes where all the numbers are odd.” Without saying another word, she took the chalk from her companion and, stepping up to the board, wrote the following:
5 x 7 + 1 35 + 1 36 ----------- = -------- = ----- = 12 3 3 3
“Great work, girls. You certainly are to be complimented, don’t we think so, everybody?” Principal Parios and the entire class, with one exception, nodded in agreement. Of course, that exception was Gamma, who had already gotten out his laptop computer, and was eagerly pressing his keyboard, entering data of some sort.
“But, alas, our pattern failed to work on the next set of three consecutive primes, namely, 5, 7, and 11,” Beta continued, with an obvious look of disappointment on her face. “Anyway, we spied another small bit of number trivia hidden in our two successes. The results of adding 1 to the products always yielded square numbers, 16 and 36.”
Sensing another “teachable moment” was at hand, Ms. Omega said to the whole class, “Perhaps if we were to take out our calculators, we could extend our search more efficiently and find another example of the pattern that Alpha and Beta have discovered for us. After all, it’s the search that’s important here, not the multiplication and division, right?”
Just then, Gamma began jumping up and down beside his desk, shouting excitedly, “Eureka! Eureka!” [That’s Greek for “I found it!”] “I’ve found another case of the pattern!”
Some kids turned to him and asked, “What is it, Gamma? What is it?”
Turning off his laptop and closing the cover, he smugly said, “Find it yourself! I’m keeping it a secret. It’s easy. At least, when you’re as smart as I am.”
Can you find Gamma’s solution? It really is easy. But I must let you in on a little secret… He has been continuing his search for a 4th solution for quite some time, with no luck. So it looks like a 4th solution just might be hard after all.
WTM would like to recognize two friends who helped make this page extra special. First, thanks go to Panagiotis Stefanides of Athens, Greece, who suggested real Greek names that I could use for the owner of the island. Second, we thank G. L. Honaker, Jr., from Bristol, VA. He created the idea of the three consecutive primes puzzle that was used in the Epilogue section. Without their kind contributions, this fairy story problem would not be as nice as it is (IMHO).