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Singapore Math Olympiad

Problem on Permutation | SMO, 2011 | Problem No. 24

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2011 based on Permutation. You may use sequential hints to solve the problem.

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2011 based on Permutation.

Permutation Problem (SMO Entrance)


A \(4 \times 4\) Sudoku grid is filled with digits so that each column , each row and each of the four \( 2 \times 2\) sub grids that composes the grid contains all of the digits from 1 to 4. For example

Sudoku - Permutation Problem
  • 288
  • 155
  • 160
  • 201

Key Concepts


Permutations & Combinations

Sudoku

Set Theory

Check the Answer


Answer: 288

Singapore Mathematical Olympiad

Challenges and Thrills – Pre – College Mathematics

Try with Hints


If you really get stuck in this problem here is the first hint to do that:

At 1st let’s consider the sub grids of \( 2 \times 2\) filled with 1-4 ( 1, 2 , 3 ,4)

If a,b,c,d are all distinct , and there are no other numbers to place in x . If {a,b} = {c,d} then again a’,b’,c,d are all distinct , and no other number can be possible for x’.

We need to understand that the choices we have ,

{a,a’} = {1,2} , {b,b’} = {3,4}, {c,c’} = {2,4} and {d,d’} = {1,3}

Among these choices \( 2^4 = 16 \) choices 4 of them are impossible – {a,b} = {c,d} = {1,4} or {2,3} and

{a,b} = {1,4} and {c,d} = {2,3} and {a,b} = {2,3} and {c,d} = {1,4}

Try rest….

Now for each remaining case a’,b’,c’ and d’ are uniquely determined so

{x} = {1,2,3,4} – {a,b} \(\cup\) {c,d}

{y} = {1,2,3,4} – {a,b} \(\cup\) {c’,d’}

{x’} = {1,2,3,4} – {a’,b’} \(\cup\) {c,d}

{y’} = {1,2,3,4} – {a’,b’} \(\cup\) {c’,d’}

In final hint :

There are 4! = 24 permutation in the left top grid we can find. So total 12 * 24 = 288 possible 4\(\times\) 4 Sudoku grids can be found.

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