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
- Smallest maximum numbers:
- Smallest magic products:
- 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?
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.
AB1. This 3x3 magic square has 7 distinct square
entries.
S2 = 541875. By Andrew Bremner.
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373˛
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289˛
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565˛
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360721
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425˛
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23˛
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205˛
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527˛
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222121
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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 54: a perfect magic
hypercube of order 5 and of dimension 4 cannot exist.
Magic squares
of polygonal numbers
Enigmas on magic
squares
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