Unsolved multimagic problems

A lot of multimagic problems are not yet solved, that's why this subject is interesting and motivating. The goal is mainly to get the maximum of characterics in the minimum of space. "What is the smallest possible xxx" and "Who will be the first to yyy" are the favorite questions.

Here is a partial list. If you have some results on an unsolved multimagic problem, send me a message! I will be pleased to add your results in this web site.

Multimagic squares using distinct integers

• What is the smallest possible bimagic square using distinct integers? The orders 3 and 4 are proved impossible, and the order 8 is the mimimum order allowing normal bimagic squares (normal = using consecutive integers). Who will be the first to construct a non-normal bimagic square of order 5, 6 or 7? (non-normal = using distinct integers). Or prove that it is impossible to construct such squares. Examples of order 6 found in February 2006 by Jaroslaw Wroblewski. Examples of order 7 found in May 2006 by Lee Morgenstern. Order 5 still unknown!
• What is the smallest possible trimagic square using distinct integers? The orders 3 and 4 are proved impossible, and the order 12 is the mimimum order allowing normal trimagic squares (normal = using consecutive integers).

Normal multimagic squares (using consecutive integers)

Multimagic cubes and hypercubes

Multiplicative magic squares

Multiplicative magic cubes

Magic squares of squares

Ten open problems from my article "Some notes on the magic squares of squares problem" published in 2005, in The Mathematical Intelligencer. Some of them were already asked above, in this page.

 373² 289² 565² 360721 425² 23² 205² 527² 222121

Problems 1 to 6: using distinct integers (the integers can be non-consecutive). Problems 7 to 10: using consecutive integers.

1. The \$100 prize offered by Martin Gardner in 1996 for nine square entries in a 3x3 magic square (or the proof of its impossibility).
2. I here offer a 100 prize + a bottle of champagne to the first who will solve an easier problem: provide a new example of a 3x3 magic square with seven distinct square entries, different than (AB1) and its rotations, symmetries, or k² multiples. Or provide any example with eight square entries. (Equivalent to Enigma 1)
3. What is the smallest bimagic square using distinct integers? Its size is unknown: 5x5, 6x6, or 7x7? My feeling is that 5x5 bimagic squares do not exist. Bimagic squares of sizes 8x8 and above are already known. Not solved, but first 6x6 bimagic squares by Jaroslaw Wroblewski in 2006, first 7x7 bimagic squares by Lee Morgenstern in 2006. 5x5 still unknown. (Equivalent to Enigma 2)
4. Construct a bimagic square using distinct prime numbers.  Solved in November 2006 by Christian Boyer, with the prime order 11. Also in June 2014 by Jaroslaw Wroblewski, with the order 8.
5. Construct the smallest possible magic square of cubes: 5a) using integers having different absolute values, 5b) using only positive integers. Not solved, but first 4x4 semi-magic square of cubes by Lee Morgenstern in June 2006. The smallest known magic square of cubes is currently a 7x7 square, by Sébastien Miquel in February 2015. 4x4, 5x5, 6x6 still unknown. (5b on 4x4 equivalent to Enigma 4, and to part 8.3 of the paper in Journal of Integer Sequences 2008, see here)
6. Construct a magic square of cubes of prime numbers. Solved in July 2007 by Jaroslaw Wroblewski and Hugo Pfoertner, with the order 42.
7. Construct the smallest possible magic cube of squares (the 16x16x16 bimagic cube, when its numbers are squared, is already a magic cube of squares)
8. Construct a bimagic cube smaller than 16x16x16.
9. Construct a perfect bimagic cube smaller than 32x32x32. Solved in April 2015 by Zhong Ming, with orders 16 and 25.
10. Demonstrate the general case of a magic (2k+1)k+1. The notation used here is coming from Richard Schroeppel : a magic (2k+1)k+1 means a perfect magic hypercube of order (2k+1) and of dimension k+1. The problem 10 is to prove for all k that -if such an object exists- its center is always the average value of the used integers. This problem is obvious for k=1, and has already been demonstrated for k=2, 3 and 4 by Richard Schroeppel. For example:
• the case k=1 means that a magic square of order 3 has always 5 in its center cell, average value of integers from 1 to 9.
• the case k=2 means that a perfect magic cube of order 5 -if it exists- has always 63 in its center cell, average value of integers from 1 to 125. Look at his demonstration. A corollary is that there is no magic 54: a perfect magic hypercube of order 5 and of dimension 4 cannot exist.

Magic squares of polygonal numbers

Enigmas on magic squares