**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)**

- Who will be the first to mathematically prove that a bimagic square of order 7 using consecutive integers is impossible? Mathematical proofs exist for the impossibility of bimagic squares of orders < 7, but today the non-existence of a bimagic square of order 7 is “only” a computing result.
- Who will be the first to construct a pandiagonal bimagic square, with all its broken lines bimagic. None is known. Or who will be the first to prove that such a square is impossible? Solved in February 2006 by Su Maoting (order 32). Hence, new problem:
- Is it possible to construct a pandiagonal bimagic square of order < 32?
- Who will be the first to construct a trimagic square of order 13 or 15? None is known. Or prove that it is impossible to construct such squares. The smallest possible trimagic square is of order 12. Trimagic squares of order 14 are impossible.
- What is the smallest possible tetramagic square? The orders < 20 are impossible, and the smallest known tetramagic square is of order 243. Who will be the first to construct a tetramagic square of order < 243?
- What is the smallest possible pentamagic square? The orders < 20 are impossible, and the smallest known pentamagic square is of order 729. Who will be the first to construct a pentamagic square of order < 729?
- What is the smallest possible hexamagic square? The orders < 20 are impossible, and the smallest known hexamagic square is of order 4096. Who will be the first to construct an hexamagic square of order < 4096?
- What are the numbers of trimagic series for squares of order 19 and above?
- What are the numbers of tetramagic series for squares of order 19 and above?
- What are the numbers of pentamagic series for squares of order 20 and above?
- What is the smallest order allowing hexamagic series? The smallest current candidates are orders 27 and 40.

**Multimagic series for squares**

**Multimagic cubes and hypercubes**

- What is the smallest possible bimagic cube? The orders < 8 are proved impossible, and the smallest known bimagic cube is of order 16. Who will be the first to construct a bimagic cube of order < 16?
- What is the smallest possible perfect bimagic cube? The orders < 8 are proved impossible, and the smallest known bimagic perfect cube is of order 32. Who will be the first to construct a perfect bimagic cube of order < 32?
- What is the smallest possible bimagic hypercube? The smallest known bimagic hypercube (dim 4 = tesseract) is of order 32. Who will be the first to construct a bimagic hypercube of order < 32?
- What are the numbers of bimagic series for cubes of order 14 and above?
- What are the numbers of trimagic series for cubes of order 11 and above?
- What is the smallest order allowing tetramagic series? The smallest current candidate is order 13.

**Multimagic series for cubes**

- Is it possible to construct additive-multiplicative magic squares of orders <= 7? Not solved, but first add-mult magic square of order 7 by Sébastien Miquel in August 2016. Orders 5 and 6 still unknowns. (Equivalent to Enigmas 6 and 6a)
- Is it possible to construct better pandiagonal multiplicative magic squares of orders >= 6?
- Construct the lists of the smallest possible products of multiplicative magic squares of orders 8 and above. Lists of orders from 3 to 7 already done.
- What are the smallest possible maximum numbers of multiplicative magic squares of orders 8 and above? Smallest possible max nbs of orders 3 to 7 already known.
- What are the smallest possible magic products of multiplicative magic squares of orders 10 and above? Smallest possible products of orders 3 to 9 already known.

**Multiplicative magic cubes**

- Smallest maximum numbers:
- What is the smallest possible maximum number of multiplicative magic cubes? And what is the order of the cube using this max nb? The smallest known max nb is 351, used in a cube of order 4.
- What is the smallest possible maximum number of perfect multiplicative magic cubes? And what is the order of the cube using this max nb? The smallest known max nb is 24992, used in a cube of order 11.
- What is the smallest possible maximum number of pandiagonal perfect multiplicative magic cubes? And what is the order of the cube using this max nb? The smallest known max nb is 24992, used in a cube of order 11.
- Smallest magic products:
- What is the smallest possible magic product of multiplicative magic cubes of order 4? The smallest known product is 4 324 320.
- What is the smallest possible magic product of perfect multiplicative magic cubes? And what is the order of the cube using this product? The smallest known product is 101 625 502 003 200 000, used in a cube of order 5.
- What is the smallest possible magic product of pandiagonal perfect multiplicative magic cubes? And what is the order of the cube using this product? The smallest known product is 89 518 183 823 250 314 294 722 560 000, used in a cube of order 8.
- And more generally (see the table):
- What are the smallest possible maximum numbers and smallest possible magic products of multiplicative magic cubes of orders 4 and above?
- What are the smallest possible maximum numbers and smallest possible magic products
of
*perfect*multiplicative magic cubes of orders 4 and above? - What are the smallest possible maximum numbers and smallest possible magic products
of
*pandiagonal perfect*multiplicative magic cubes of orders 4 and above?

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.

373˛ |
289˛ |
565˛ |

360721 |
425˛ |
23˛ |

205˛ |
527˛ |
222121 |

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

- The $100 prize offered by Martin Gardner in 1996 for nine square entries in a 3x3 magic square (or the proof of its impossibility).
- 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 k˛ multiples. Or provide any example with eight square entries. (Equivalent to Enigma 1) - 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)
- Construct a bimagic square using distinct prime numbers.
^{[9] [50]}Solved in November 2006 by Christian Boyer, with the prime order 11. Also in June 2014 by Jaroslaw Wroblewski, with the order 8. - 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) - Construct a magic square of cubes of prime numbers.
^{[9]}Solved in July 2007 by Jaroslaw Wroblewski and Hugo Pfoertner, with the order 42. - Construct the smallest possible magic cube of squares (the 16x16x16 bimagic cube, when its numbers are squared, is already a magic cube of squares)
- Construct a bimagic cube smaller than 16x16x16.
- Construct a perfect bimagic cube smaller than 32x32x32. Solved in April 2015 by Zhong Ming, with orders 16 and 25.
- 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 5
^{4}: a perfect magic hypercube of order 5 and of dimension 4 cannot exist.

**Magic squares
of polygonal numbers**

- Who will construct a 3x3 magic square of distinct triangular numbers, or its equivalent 3x3 magic square of squares? Or who will prove that it is impossible?
- What is the smallest possible magic squares of distinct pentagonal numbers: 3x3, 4x4 or 5x5? Not solved, but first 4x4 and 5x5 squares by Lee Morgenstern in November 2007. 3x3 still unknown.

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