The Steffi problem concerns integer ratios between numbers obtained by permuting the same set of digits.
A homework problem proposed in Steffi's math class in January 2003 asked students
to prove that no ratio of two unequal numbers obtained
by permuting all the digits 1, 2, ..., 7 results
in an integer. If the numerator 
and denominator 
satisfied 
for an integer 
, then 
, since the numbers are positive and unequal and
. By the
divisibility test for 9, every such permutation
is congruent to 
, since the sum of its digits
is 28. Hence 
implies 
,
which is impossible for 
.
In general, for integers 
and 
, consider the number of ordered
pairs 
of unequal numbers obtained by permuting all the digits
1, 2, ..., 
in base 
, whose ratio 
is an integer. Then there is
a unique 
solution
 |
(1)
|
a unique 
solution
 |
(2)
|
three 
solutions
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
and so on.
The number of solutions for the first few bases and numbers of digits 
are summarized in the table below (OEIS A080202).
 | # solutions for digits  ,  , ...,  |
| 3 | 0 |
| 4 | 0, 1 |
| 5 | 0, 0, 1 |
| 6 | 0, 0, 3, 25 |
| 7 | 0, 0, 0, 2, 7 |
| 8 | 0, 0, 0, 0, 68, 623 |
| 9 | 0, 0, 0, 0, 0, 124, 1183 |
| 10 | 0, 0, 0, 0, 0, 0, 2338, 24603 |
| 11 | 0, 0, 0, 0, 0, 0, 3, 598, 5895 |
| 12 | 0, 0, 0, 0, 0, 0, 0, 0, 161947, 2017603 |
As can be seen from the table, in base 10, the only solutions are for the digits 12345678 and 123456789. Of the solutions
for 
,
there are two that produce three different integers for
the same numerator:
 |  |  |
(6)
|
 |  |  |
(7)
|
Taking the diagonal entries 
from this list for 
, 4, ... gives the sequence 0, 1, 1, 25, 7, 623, 1183, 24603,
... (OEIS A080203).
