## A Multiplication Based Logic Puzzle

### 426 My Response to a Pi-lish Question

A comma is used for decimals in many countries.

This last week there was a post on the Mathemagical Site titled “Do You Speak Pilish?”  Some people remember the digits of π by memorizing carefully constructed sentences in which the first word has three letters, the second word has one letter, and so on. Several examples were given, not just in English, but in eight other languages as well!

Hungarian was not one of the languages listed, but I wondered if there could possibly be a Pilish way for Hungarians to remember the digits of pi? (Almost all of my husband’s relatives were born in Hungary, and I am fascinated with the country and the language.) I just had to google “Magyar pi szám,” to find an article titled Minden idők legjobb magyar nyelvű pi-verse.

Now while I can read many Hungarian words, the sentence structure is so different from English that my comprehension isn’t as good as I’d like it to be. My son, David, taught himself the basics of the language before he went there to live and work several years ago. I emailed him the article requesting that he help me with the translation. In the email he sent back you will notice the problem with word for word translation of Hungarian into English. My son wrote:

“I don’t think I could translate it whilst maintaining the word lengths (which is the whole point). I’m giving it to you with a more or less word for word translation along with one that is written in more natural English. The Ludolph it mentions in the poem is the Dutch mathematician Ludolph van Ceulen, who was the first to publish pi up to 20 digits.”

I put his word for word translation in the following graphic:

Here is David’s translation into more natural English:

• Instead of the old and rough approximation,
• Count the letters that come, word for word
• If we end here at twenty words, we already have Ludolph’s result,
• but exactly 10 more come from this last stanza.
• That, I can promise confidently.”

Here is my answer to the question, “Do you speak Pilish?”

Not really. I am not the least bit interested in memorizing some cute paragraph in English to help me remember the first 30 or so digits of pi, BUT in Hungarian, I am going to give a try!

• 426 is a composite number.
• Prime factorization: 426 = 2 x 3 x 71
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 426 has exactly 8 factors.
• Factors of 426: 1, 2, 3, 6, 71, 142, 213, 426
• Factor pairs: 426 = 1 x 426, 2 x 213, 3 x 142, or 6 x 71
• 426 has no square factors that allow its square root to be simplified. √426 ≈ 20.6398

### 425 and Level 6

425 ends in 25 so it can be divided evenly by 25. If I had \$4.25 all in quarters. How many quarters would I have? That’s the problem that I think of when I divide by 25. All of the factors of 425 are listed below the puzzle.

For some reason unknown to me, here in the United States, dates are ordered by month, date, and year. This rather illogical way of ordering allows us to say that today is 3-14-15, which are the first five digits of pi.  It could also be said that 3-14-15 at 9:26:53 gives the first ten digits of pi.

Logical or not, it is fun to declare today as Pi Day. Today’s puzzle celebrates those first five digits:

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

• 425 is a composite number.
• Prime factorization: 425 = 5 x 5 x 17, which can be written 425 = (5^2) x 17
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 425 has exactly 6 factors.
• Factors of 425: 1, 5, 17, 25, 85, 425
• Factor pairs: 425 = 1 x 425, 5 x 85, or 17 x 25
• Taking the factor pair with the largest square number factor, we get √425 = (√25)(√17) = 5√17 ≈ 20.6155

425 and all of it factors (except 1) are hypotenuses of primitive Pythagorean triples, so 425 is the hypotenuse of several triples:

• [87-416-425] and
• [297-304-425] are primitives
• [65-420-425] is [13-84-85] times 5
• [119-408-425] is [7-24-25] times 17
• [180-385-425] is [36-77-85] times 5
• [200-375-425] is [8-15-17] times 25
• [255-340-425] is [3-4-5] times 85