![magic calculator emoji ti99 magic calculator emoji ti99](https://commondatastorage.googleapis.com/images.pricecharting.com/AMIfv96HGV0QJ-3U7N6zloZikj4y22dY1H4VcTnXsI9gUq9PBTN_cKZf1e34s2W915iIXSbuuhgoC21iDGQkb_TuXrzLlfPu2mMPX1nwOC0PbOIV-B6JqSGse8k9MgWUKeC6OOV7YYSMwzKmDiptuFeO0sbEfxIGPA/240.jpg)
Now let's work out a systematic way of finding all $3\times 3$ magic squares. In particular, it seems that the magic number is always a multiple of $3$. Each time you succeed in making a magic square, you should check that the magic product and magic pairwise product properties also work.Īfter having worked out a few more examples, you may notice that the magic number is always three times the middle number of the magic square. Try to make up your own magic squares by assigning a few numbers at random to some of the cells and then filling the other cells in so that the rows, columns and diagonals add up to the same number. In this case it is not at all clear that the new magic square will still have the magic product and the magic pairwise product properties. You could also add the same number to each of the nine numbers in a magic square, and the result would clearly be a new magic square. The new squares would still be magic, and it is not difficult to see that the magic squares you create would again have the magic product property and the magic pairwise product property. Of course, you could just rotate and reflect the examples above, and you could multiply every number in the square by the same constant. Magic number 24 magic product 840 magic pairwise product 489Ĭan you find other $3\times 3$ magic squares? Magic number 18 magic product 414 magic pairwise product 285 6 Magic number 18 magic product 468 magic pairwise product 294 4 However, magic squares can be formed using different sets of nine numbers.
![magic calculator emoji ti99 magic calculator emoji ti99](https://elefunstore.ro/13909-large_default/balon-mini-corp-unicorn-magic-pink-happy-birthday.jpg)
These examples all have the same magic sum, magic product and magic pairwise product as the first one. The only other $3\times 3$ magic squares using the numbers from $1$ to $9$ are just reflections and rotations of the magic square above, such as 8 $(8\times 3+3\times 4+4\times 8)+(1\times 5+5\times 9+9\times 1)+ (6\times 7+7\times 2+2\times 6)=195$.Īgain the results are the same! This number is called the magic pairwise product of the square. Now take the "pairwise products'' in each row and add them all up: This number is called the magic product of the square. You get the same total as when you multiply the three numbers in each column together and add the three products: Create an expressive cartoon avatar, choose from a growing library of moods and stickers - featuring YOU Put them into any text message, chat or status update. It is less well known that if you multiply the three numbers in each row together and add the three products: