Answers to last issue's Brain Teasers
1. Consider the string of six numbers from 1 to 6. Can you place them in the circles set in a triangular pattern so that the numbers on each side add up to the same sum?
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Now take the numbers from 1 to 9. Can you place them in the larger triangular pattern so that each side again sums up to the same number?
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Answer: There are several answers, as you would see if you had tried it out yourself. For example, two possible answers which give the smallest and largest sums for the first case are shown in the figure. Each side adds up to 9 or 12. A possible solution for the second case is also shown in the figure, where the sum on each side is 20.
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2. Sudoku
You must all have seen the popular game of Sudoku. There are 9 rows and 9 columns with 81 little squares in all. Some of the squares are filled. The aim of the game is to fill in the blank squares with numbers from 1 to 9 so that no number repeats in either a row or a column or a sub-square. A sub-square is a collection of 9 little squares with three rows and three columns so there are 9 sub-squares in the big Sudoku square.
Let us think of the simpler problem of a 4 X 4 Sudoku. As shown in the figure, there are 4 rows, 4 columns and 4 sub-squares which are of size 2 by 2 each (as denoted by the dark lines). Here the aim is to fill in the numbers from 1 to 4 with no repeats in row, column or sub-square.
The problem: One sub-square of the Sudoku is correctly filled, with no other entries. Can you correctly complete the Sudoku according to the rules? What about the case of a 9 X 9 Sudoku?
Answer: It can be filled correctly. Start with the top row. The entries are 1,2 (as boxed in the figure). The remaining numbers 3,4 can therefore be simply filled in (either as 3,4 or 4,3) in the top row.
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Similarly, the entries 3,4 are boxed in the figure. The remaining entries of that row must be 1,2. Simply fill it in. The top two rows (and sub-squares) are complete. We have to fill in the bottom 2 sub-squares.
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Now, consider the boxed numbers 1,3 in the first column (see the next figure). The remaining numbers are 2,4 and we fill it in any order in the remaining boxes of the first column. Similarly, we fill 1,3 in the second column. We are left with the last sub-square (or the last two rows).
In the third row, we already have 2,1. So we have to fill 3,4. Now, we cannot put either number in either box. That is because 3,4 are already there in the top row. See the boxed numbers 3,4 in the top right sub-square of the next figure.
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So, you can only put 4,3 in the third row. Similarly, you can put 2,1 in the last row to complete the Sudoku, as shown in the figure. No row or column has repeated entries, neither does the sub-square.
An extension of this argument also works for a 9X9 Sudoku.