Problems

Filters
Clear Filters

1 problem found

1992 Paper 1 Q2
D: 1500.0 B: 1500.0
combinatorics magic square proof enumeration constraint satisfaction systematic counting

A \(3\times3\) magic square is a \(3\times3\) array \[ \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h & k \end{array} \] whose entries are the nine distinct integers \(1,2,3,4,5,6,7,8,9\) and which has the property that all its rows, columns and main diagonals add up to the same number \(n\). (Thus \(a+b+c=d+e+f=g+h+k=a+d+g=b+e+h=c+f+k=a+e+k=c+e+g=n.)\)

  1. Show that \(n=15.\)
  2. Show that \(e=5.\)
  3. Show that one of \(b,d,h\) or \(f\) must have value \(9\).
  4. Find all \(3\times3\) magic squares with \(b=9.\)
  5. How many different \(3\times3\) magic squares are there? Why?
{[}Two magic squares are different if they have different entries in any place of the array.{]}

Solution:

  1. \((a+b+c)+(d+e+f)+(g+h+k) = 3n = 1 + 2 + \cdots + 9 = 45 \Rightarrow n = 15\).
  2. Summing all rows, columns, diagonals through \(e\) we have \((a+e+k)+(b+e+h)+(c+e+g)+(d+e+f) = 45 + 3e = 60 \Rightarrow e = 5\).
  3. Suppose that one of the corners is \(9\), then we need to find \(2\) ways to make \(6\) not using \(5\) and \(1\) (as \(5\) is in the middle and \(1\) diagonally opposite). Clearly this is not possible as the only remaining numbers are \(2,3,4\) and only \(2+4 = 6\). Therefore \(9\) cannot be in the corner or central squares, ie it's one of \(b,d,h,f\)
  4. We must have \begin{array}{ccc} a & 9 & c\\ d & 5 & f\\ g & 1 & k \end{array} and so \(a\) or \(c = 4\). Once we place \(4\) by symmetry there will be another solution with \(a = 2\). So: \begin{array}{ccc} 4 & 9 & 2\\ d & 5 & f\\ g & 1 & k \end{array} we now see \(k\), then \(f\), then \(d\) then \(g\) must be determined, ie: \begin{array}{ccc} 4 & 9 & 2\\ 3 & 5 & 7\\ 8 & 1 & 6 \end{array} so our two solutions must be this and \begin{array}{ccc} 2 & 9 & 4\\ 7 & 5 & 3\\ 6 & 1 & 8 \end{array}
  5. For each of the \(4\) possible placements of \(9\) there are two magic squares, so there are \(8\) possible magic squares, all related by reflection and rotation.

View