Category: Magic Squares

Magic Squares With Perfect Square Sum of Entries: Orders 3 to 31

This work shows how to create magic squares with a perfect square number for the total sum entries. This has been done in five ways. Initially, the two ways are using entries as consecutive odd numbers and consecutive natural numbers for odd order magic squares and consecutive fraction numbers for even order magic squares. This … Continue reading Magic Squares With Perfect Square Sum of Entries: Orders 3 to 31

Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 47

There are many ways of writing magic or bordered magic squares, where the sum of entries is always a perfect square. In one of the possibility, the magic sums are such that they satisfy uniformity property. Another way is to write magic squares generated by Pythagorean triples. Based on these idea, we can always write … Continue reading Minimum Perfect Square Sum Bordered and Block-Wise Bordered Magic Squares: Orders 3 to 47

Area Representations Magic Squares With Fraction Numbers Entries

It is well known that every magic square can be written as perfect square sum of entries. It is always possible with odd number entries starting from 1. In case of odd order magic squares we can also write with consecutive natural number entries. Still, it is unknown whether it is possible to even order … Continue reading Area Representations Magic Squares With Fraction Numbers Entries

Bordered and Block-Wise Bordered Magic Squares: Odd Order Multiples

In the previous work we did block-wise bordered and block-wise block-bordered magic squares mutiples of even orders. See the link below: Inder J. Taneja, Bordered and Block-Wise Bordered Magic Squares: Even Order Multiples, https://inderjtaneja.com/2021/02/12/bordered-and-block-wise-bordered-magic-squares-even-order-multiples/ The even order multiples considered are of orders 6, 8, 10, 12, 14, 16, 18 and 22. In this case, we have … Continue reading Bordered and Block-Wise Bordered Magic Squares: Odd Order Multiples

Bordered and Block-Wise Bordered Magic Squares: Even Order Multiples

It is always possible to block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand, one can always write bordered magic squares of any order. In the previous works, the author combined the both, i.e., bordered and block-wise magic squares, … Continue reading Bordered and Block-Wise Bordered Magic Squares: Even Order Multiples

Block-Wise and Block-Bordered Magic and Bimagic Squares of Orders 10 to 47

We know that we can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine the both, i.e., bordered and … Continue reading Block-Wise and Block-Bordered Magic and Bimagic Squares of Orders 10 to 47

2-Digits Universal Magic Squares Up To Orders 128

The whole work is without use of any programming language.It is just based on the number's combinations. Part I: Orders 128, 126 and 120 This work brings magic squares multiples of 4, 6 and 12 using only two digits: (1,8), (2,5) and (6,9). Each magic square contains 14-digits in each cell. Just with two digits, … Continue reading 2-Digits Universal Magic Squares Up To Orders 128

Block-Wise and Block-Bordered Magic and Bimagic Squares With Magic Sums 21, 21^2 and 2021

We can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to bring both bordered and block-wise magic squares. These block-wise … Continue reading Block-Wise and Block-Bordered Magic and Bimagic Squares With Magic Sums 21, 21^2 and 2021

Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2021

The idea of bordered magic squares is well known in the literature. In this work, bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2021. The work is for the orders 3 to 26. The work include fractional and decimal numbers entries having positive … Continue reading Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2021

Block-Bordered Magic Squares of Prime and Double Prime Orders: I, II and III

We can always write block-wise magic squares of any order except for the orders of type p and 2p, where p is a prime number. On the other hand we can always write bordered magic squares of any order. The aims of this work is to combine bordered and block-wise magic squares, for the magic … Continue reading Block-Bordered Magic Squares of Prime and Double Prime Orders: I, II and III

Upside-Down Magic and Bimagic Squares of Orders 3 to 32 Using 6 and 9

This work brings, magic squares of order 3 to 32 just with two digits in such a way that the magic squares are upside-down independent of magic sums. This is done only with two digits 6 and 9. The works for the digits 1 and 8, and 2 and 5 are given separately. In case of 2 and 5, the … Continue reading Upside-Down Magic and Bimagic Squares of Orders 3 to 32 Using 6 and 9

Universal Magic and Bimagic Squares of Orders 3 to 32 Using 2 and 5

This work brings, magic squares of order 3 to 32 just with two digits in such a way that the magic squares are upside-down and mirror looking independent of magic sums. These types of magic squares we call as universal magic squares. This is done only with two digits 2 and 5. In this case the numbers 2 and 5 are written in digital form. … Continue reading Universal Magic and Bimagic Squares of Orders 3 to 32 Using 2 and 5

Universal Magic and Bimagic Squares of Orders 3 to 32 Using 1 and 8

This work brings, magic squares of order 3 to 32 just with two digits in such a way that the magic squares are upside-down and mirror looking independent of magic sums. These types of magic squares we call as universal magic squares. This is done only with two digits 1 and 8. The work for … Continue reading Universal Magic and Bimagic Squares of Orders 3 to 32 Using 1 and 8

Universal Palindromic Day (02.02.2020) and Two Digits Magic Squares

Today,  the February 02, 2020 is the universal palindromic day. It can be written as 02.02.2020 or 2020.02.02 Some idea on this day can be seen in Alex Bollos column in "The Guardian": Alex Bellows, https://www.theguardian.com/science/2020/jan/27/can-you-solve-it-toot-toot-for-world-palindrome-day., January 27, 2020 The palimdromic magic square appeared in the above column: Still there are two more days in … Continue reading Universal Palindromic Day (02.02.2020) and Two Digits Magic Squares

Bordered Magic Squares With Order Square Magic Sums

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. In this work, nested magic squares are studied in such a way that the magic sums are equal to the order of the magic square. The study include integer values. In some cases decimal entries with positive … Continue reading Bordered Magic Squares With Order Square Magic Sums

Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2020

The idea of bordered magic squares is well known in the literature. In this work, bordered magic squares are constructed in such a way that the final magic sum of each bordered magic square is 2020. The work is for the orders 3 to 25. In each case, a symmetric result is found for the … Continue reading Fractional and Decimal Type Bordered Magic Squares With Magic Sum 2020

Symmetric Properties of Nested Magic Squares

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. In this work, nested magic squares are studied for the consecutive natural numbers for the orders 5 to 25. Properties like, sub-magic squares sums, total entries sums, borders entries sums, etc. are studies. Final results lead us to symmetric properties. … Continue reading Symmetric Properties of Nested Magic Squares

Nested Magic Squares With Perfect Square Sums, Pythagorean Triples, and Borders Differences

The idea of nested magic squares is well known in the literature, generally known by bordered magic squares. Instead, considering entries as consecutive numbers, we considered consecutive odd numbers entries. This gives us perfect square sum magic squares. We worked with magic squares of orders 3 to 25. Finally, the result can be written in … Continue reading Nested Magic Squares With Perfect Square Sums, Pythagorean Triples, and Borders Differences