Using the sixteen two-digit numbers made from the digits 1, 6, 8 and 9 (11, 16, 18, 19, 61, 66, 68, 69, 81,...
arithmetic/upside.down.magic.square New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Using the sixteen two-digit numbers made from the digits 1, 6, 8 and 9 (11, 16, 18, 19, 61, 66, 68, 69, 81, 86, 88, 89, 91, 96, 98, 99), construct a 4x4 magic square that is still magic when turned upside-down.
Show solution
11 89 98 66
96 68 19 81
69 91 86 18
88 16 61 99
Discounting rotations and reflections, how many different
magic squares can be made with this set of numbers?
-- Stephen H. Landrum <slandrum@ntg.com>
[ Editor: asked how he found it, Landrum replied: ]
I did it by hand.
It's pretty easy when you realize that every row/column/diagonal must
have one of each of 1,6,8,9 and one of each of 10,60,80,90.
First, make a magic square out of the set
{1,1,1,1,6,6,6,6,8,8,8,8,9,9,9,9}:
1 6 8 9
9 8 6 1
6 1 9 8
8 9 1 6
Make a second square by multiplying every cell in the first by
10, and rotate the square by 90 degrees or flip it along one of
the diagonals. Add the two squares together and you have a
solution to the problem.
This is just a Graeco-Roman square of order 4 with additional
restrictions on the diagonals.
-- Stephen H. Landrum <slandrum@ntg.com>
[ Editor: the puzzle was posted by Johan Myrberger
<myrberger@e.kth.se>. Landrum's question about the number of such
squares was not answered in the collected postings. ]