15x15 magic squares of cubes.
15x15 magic squares of 4th powers.
16x16 magic squares of 4th powers.
16x16 magic squares of 5th powers.
25x25 magic squares of 5th powers.
See also the Magic squares of cubes general page


15x15 magic squares of cubes
15x15 magic squares of 4th powers

15x15 magic squares of consecutive cubes should be possible but are still unknown. In order to construct a square of non-consecutive cubes, we can use a method similar to Morgenstern's 6x6 method. With this taxicab(3, 3, 5) number:

and with this taxicab(3, 5, 8) number, but without using [2][5][6][7][8]:

we can construct this 15x15 semi-magic square of magic sum S3 = 4053232 * 14697, with MaxNb = (150 * 22)^3. Using the taxicab method, this is the smallest possible solution (smallest S3, smallest MaxNb) producing 15*15 = 225 distinct integers. This square can be downloaded: see at the end of this page. Here is the square, with cells rearranged in order to have one magic diagonal, two diagonals being unfortunately impossible. Are two diagonals possible, perhaps with a different square using another combination of taxicab numbers?


What about 4th powers? Using a similar method, with this taxicab(4, 3, 6) number without using [5]:

and with this taxicab(4, 5, 4) number, but without using [2] :

we can construct this 15x15 semi-magic square of magic sum S4 = 292965218 * 794179, with MaxNb = (127 * 29)^4. Using the taxicab method, this is the smallest possible solution (smallest S4, smallest MaxNb) producing 15*15 = 225 distinct integers. This square can be downloaded: see at the end of this page.

François Labelle in 2010

In September 2011, rearranging the cells of my square above, François Labelle succeeded in obtaining two magic diagonals! This is the new smallest known magic square of 4th powers, coming after his previous 16x16 record. François Labelle, PhD Berkeley, is a Canadian working at Google USA. His homepage is http://wismuth.com

This square can be downloaded: see at the end of this page. More details on this work: http://wismuth.com/magic/squares-of-nth-powers.html


16x16 magic squares of 4th powers
16x16 magic squares of 5th powers 

A 16x16 magic square of cubes is obtained from Chen Mutian - Chen Qinwu's 16x16 trimagic square, by cubing its integers. This square of cubes, constructed in 2005, uses consecutive cubed integers. But 16x16 magic squares of 4th powers were unknown, using consecutive or non-consecutive 4th powers. Up to the end of 2010, the smallest known magic square of 4th powers was bigger, coming from Li Wen's 36x36 pentamagic square raising its integers to the 4th power.

In December 2010, after his 12x12 semi-magic square of 4th powers, François Labelle constructed a 16x16 magic square: this was the smallest known magic square of 4th powers before the above 15x15 square.

In order to obtain a semi-magic square, inspired by Morgenstern's 4x4 method, he used this taxicab(4, 4, 4) number:

and this taxicab(4, 4, 6) number, but without using [3] and [6]:

Using the taxicab method, this is the smallest possible solution (smallest S4, smallest MaxNb) producing 16*16 = 256 distinct integers.Then, he rearranged the cells in order to get two magic diagonals (a difficult task!), producing the following magic square with S4 = 1950354 * 321793923, and MaxNb = (37 * 123)^4. This square can be downloaded: see at the end of this page.

In March 2013, Toshihiro Shirakawa found a new construction method for (2m)4th-order magic squares of nth powers. With m=1, giving the order 16, if

then this square, when raising its cells at the nth power, is magic

Using the smallest four taxicab(4,2,2):

Toshiriro obtained a 16x16 magic square of 4th powers having:

This square, with bigger numbers than Labelle's square above, can be downloaded: see at the end of this page.


What about 16x16 magic squares of 5th powers? In June 2011, Jaroslaw Wroblewski searched for taxicab(5, 4, 4) numbers, and found only this one example using integers < 4000^5:

Because we need two taxicab(5, 4, 4), the idea is to try to "enhance" one of the numerous taxicab(5, 4, 3), multiplying by k^6, with small k.
Jaroslaw found 4124 taxicab(5, 4, 3) with integers < 4000^5. Using the 388th:

and multiplying it by 6^5, a new decomposition appears, giving this new taxicab(5, 4, 4):

Because these two taxicab(5, 4, 4) generate 256 distinct integers, we now have a semi-magic square of 5th powers! This square can be downloaded: see at the end of this page. François Labelle also obtained in June 2011 the second taxicab number, using the same method, but did not obtain the first one (~1.07e+17) because his search was at that time for less than 6e+16.


25x25 magic squares of 5th powers

The smallest known magic square of 5th powers is 36x36, coming from Li Wen's 36x36 pentamagic square and raising its integers to the 5th power.

In order to beat this record and to construct a smaller magic square of 5th powers, one method is to produce two taxicab(5, 5, 5) numbers. With this method, it is extremely difficult to obtain 25*25 = 625 distinct integers, but here is a solution! If I am right, this is the only solution using two taxicab numbers < 1.03e+13.

The semi-magic square produced by these two numbers has S5 = 8086892107975 * 10235797246718, and MaxNb = (367 * 392)^5.

This square can be downloaded: see at the end of this page.
Is it possible to rearrange its cells, obtaining two magic diagonals? It could become the new smallest known magic square of 5th powers!

In May 2011, François Labelle computed taxicab numbers up to 5.56e+13, and found these two numbers:

They generate a semi-magic square with 625 distinct integers, with the smallest possible S5 = 4.75e+25 and the smallest possible MaxNb = 125715^5. Better than my above solution S5 = 8.28e+25 and MaxNb = 143864^5.


Download squares of this webpage


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