VII.

MAGICAL SQUARES WITH MAGICAL PARTS.

If we divide a square of 8 times 8 places by means of the two middle lines parallel to its sides into 4 parts containing each 4 times 4 spaces, we may propound the problem of so inserting the numbers from 1 to 64 in these spaces that not only the whole shall form a magic square, but also that each of the 4 parts individually shall be magical, that is to say, give the same sum for each row. This problem also has been successfully solved, as the following diagram will show.

[Illustration: Fig. 27.

+--+--+--+--++--+--+--+--+ | 1| 4|63|62|| 5| 8|59|58| +--+--+--+--++--+--+--+--+ |64|61| 2| 3||60|57| 6| 7| +--+--+--+--++--+--+--+--+ |42|43|24|21||34|35|32|29| +--+--+--+--++--+--+--+--+ |23|22|41|44||31|30|33|36| +===========++===========+ |13|16|51|50|| 9|12|55|54| +--+--+--+--++--+--+--+--+ |52|49|14|15||56|53|10|11| +--+--+--+--++--+--+--+--+ |38|39|28|25||46|47|20|17| +--+--+--+--++--+--+--+--+ |27|26|37|40||19|18|45|48| +--+--+--+--++--+--+--+--+ ]

The 4 numbers in each row of any one of the sub-squares here, gives 130; so that the sum of each one of the rows of the large square will be 260.

Finally, in further illustration of this idea, we will submit to the consideration of our readers a very remarkable square of the numbers from 1 to 81. This square, which will be found on the following page (Fig. 28), is divided by parallel lines into 9 parts, of which each contains 9 consecutive numbers that severally make up a magic square by themselves.

[Illustration: Fig. 28.

+---+---+---++---+---+---++---+---+---+ | 31| 36| 29|| 76| 81| 74|| 13| 18| 11| +---+---+---++---+---+---++---+---+---+ | 30| 32| 34|| 75| 77| 79|| 12| 14| 16| +---+---+---++---+---+---++---+---+---+ | 35| 28| 33|| 80| 73| 78|| 17| 10| 15| +===========++===========++===========+ | 22| 27| 20|| 40| 45| 38|| 58| 63| 56| +---+---+---++---+---+---++---+---+---+ | 21| 23| 25|| 39| 41| 43|| 57| 59| 61| +---+---+---++---+---+---++---+---+---+ | 26| 19| 24|| 44| 37| 42|| 62| 55| 60| +===========++===========++===========+ | 67| 72| 65|| 4 | 9 | 2 || 49| 54| 47| +---+---+---++---+---+---++---+---+---+ | 66| 68| 70|| 3 | 5 | 7 || 48| 50| 52| +---+---+---++---+---+---++---+---+---+ | 71| 64| 69|| 8 | 1 | 6 || 53| 46| 51| +---+---+---++---+---+---++---+---+---+ ]

Wonderful as the properties of this square may appear, the law by which the author constructed it is equally simple. We have simply to regard the 9 parts as the 9 cells of a magic square of the numbers from I to IX and then to inscribe by the magic prescript in the square designated as I the numbers from 1 to 9, in the square designated as II the numbers from 10 to 18, and so on. In this way the square above given is obtained from the following base-square:

[Illustration: Fig. 29.

+----+----+----+ | IV | IX | II | +----+----+----+ | III| V | VII| +----+----+----+ |VIII| I | VI | +----+----+----+ ]