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CAT 2017 Slot 2 — QA Question 1

MiscellaneousEasy

The numbers 1,2, …, 9 are arranged in a 3 × 3 square grid in a such way that each number occurs once and the entries along each column, each row, and each of the two diagonals add up to the same value.

If the top left and the top right entries of the grid are 6 and 2, respectively, then the bottom middle entry is

Answer & solution

Correct answer: 3

Answer: 3

Solution

Easy

This is a 3×33\times3 magic square using 11–99. The constant sum of every row, column and diagonal is fixed, so once we know it we can fill the missing cells one subtraction at a time.

6 2 7 5 3
1

Find the magic constant. The nine entries are 11 through 99, and the three rows together cover every entry exactly once.

1+2+⋯+9=45⇒ row sum=453=15(three equal rows)\begin{aligned} &1+2+\cdots+9=45\\ &\Rightarrow\ \text{row sum}=\frac{45}{3}=15 \quad\text{(three equal rows)} \end{aligned}
2

Top-middle cell. The top row already has 66 and 22, so the remaining top entry completes the row to 1515.

15−6−2=7\begin{aligned} &15-6-2=7 \end{aligned}
3

Centre cell. In a 3×33\times3 magic square the centre is always the average of all entries, since the two diagonals, the middle row and the middle column all pass through it.

centre=459=5\begin{aligned} &\text{centre}=\frac{45}{9}=5 \end{aligned}
4

Bottom-middle cell. The middle column is 77 (top, step 2), 55 (centre, step 3) and the unknown bottom entry, summing to 1515.

15−7−5=3\begin{aligned} &15-7-5=3 \end{aligned}
Bottom middle entry=3\text{Bottom middle entry}=3

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CAT 2017 Slot 2 QA Q1: The numbers 1,2, …, 9 are arranged in a 3 × 3 square grid in a such way that each number occurs o — Solution | TheCATExam