An alphamagic square is a magic square that remains magical when its numbers are replaced by the number of letters in the name of each of the numbers. Thus, for the English language, the number 3 is replaced by the number 5, the number of letters in the English word "three" ( Russian three ). Since the number of letters for each number will be different in different languages, the alpha squares depend on the language used. [1] The Alphamagic Squares were invented by Lee Sallows in 1986. [2] [3]
Content
English example
The example below is the alpha square. To make sure that some kind of magic square is alpha-magic, you need to translate it into a form in which all numbers are turned into their names. For example:
| five | 22 | 18 |
| 28 | 15 | 2 |
| 12 | eight | 25 |
turns into
| five | twenty-two | eighteen |
| twenty-eight | fifteen | two |
| twelve | eight | twenty-five |
Counting the number of letters in each number creates a square, which also turns out to be magic:
| four | 9 | eight |
| eleven | 7 | 3 |
| 6 | five | ten |
If the generated square is magic, the original square is called alpha-magic. In 2017, British computer scientist Chris Patuzzo discovered several double alpha squares in which the alpha square generated from the original square is also alpha (i.e. the original square is alpha twice). [four]
The square above has another special property: if you take all nine numbers from them and put them in ascending order, then each of them will follow the previous one. This fact prompted Martin Gardner to describe the square as "By far the most excellent magic square that has ever been found." (English "Surely the most fantastic magic square ever discovered." [5] ).
Other languages
In 2018, Jamal Senjaya found the first 3 × 3 alpha-mosaic square for the Russian language (see below). Further, the same person found another 158 alpha-magic squares 3 × 3 in size for the Russian language, in which not a single number exceeds 300.
The Universal Book of Mathematics provides the following information about alpha squares: [6] [7]
Surprisingly, a large number of 3 × 3 alpha squares exist for English and other languages. For the French language, there is only one 3 × 3 alfamagic square in which the numbers do not exceed 200, but there are another 255 squares if each of the numbers does not exceed 300. For numbers less than 100, no such squares exist for Danish or Latin, but there are 6 for the Dutch, 13 for the Finnish, and an incredible amount - 221 for the German. Until now, a 3 × 3 magic square has not yet been found, from which one could create another magic square, and then create another magic square from the resulting square (the so-called "Alpha magic triple"). Also, not a single alpha square of sizes 4 × 4, 5 × 5 or more is unknown for any of the languages.
Example for the Russian language
Below is the first alfamagic square found for the Russian language in size 3 × 3.
| 119 | 213 | sixteen |
| 13 | 116 | 219 |
| 216 | nineteen | 113 |
which in turn turns into
| one hundred and nineteen | two hundred and thirteen | sixteen |
| thirteen | one hundred sixteen | two hundred nineteen |
| two hundred and sixteen | nineteen | one hundred thirteen |
Counting the number of letters in each of the nine cells, a square is created that is also magic:
| 15 | sixteen | eleven |
| ten | 14 | 18 |
| 17 | 12 | 13 |
One more interesting property should be noted: if we take all the numbers of the generated square in ascending order, then each of them strictly follows each other (starting with the number 10 and ending with 18).
Notes
- ↑ Wolfram MathWorld: Alphamagic Squares
- ↑ Mathematical Recreations: Alphamagic Square by Ian Stewart , Scientific American:, January 1997, pp. 106-110
- ↑ ACM Digital Library, Volume 4 Issue 1, Fall 1986
- ↑ Double Alphamagic Squares Futility Closet , November 16, 2015
- ↑ Gardner, Martin (1968), A Gardner's Workout: Training the Mind and Entertaining the Spirit, p. 161, AK Peters / CRC Press, Natick, Mass., July 2001, ISBN 1568811209
- ↑ The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes , by David Darling, p. 12, Hoboken, NJ: Wiley, 2004 , ISBN 0471270474
- ↑ Encyclopedia of Science, Games & Puzzles: Alphamagic Squares