#Puzzle 35.1

The Catwoman rang the bell. “Here are my cats,” she announced as her neighbour let her in.

The two women and the cats trooped inside. “Let’s begin with the kitchen. It has twice as many mice as the bedroom,” the house owner said.

“ ₹5,000 for a day’s work, no matter how many cats or mice are involved,” the Catwoman said.

The day’s work began. The cats set about exterminating the mice in the kitchen. Half the day eventually passed.

“Let us now divide the workforce,” the Catwoman offered, and took half her cats to the bedroom.

The two equal teams now worked separately. At precisely the moment when the workday ended (at sunset), the last of the kitchen mice was taken care of. In the bedroom, however, some mice remained when the team left.

The Catwoman arrived the next morning, this time with only one cat and one reminder: “Same ₹5,000 for a day’s work, no matter how many cats or mice.”

The solo cat finished off the bedroom’s remaining mice precisely at sunset. The bill was settled at ₹10,000.

If each cat killed mice at the same rate, how many cats were in the original team?

Different solvers have used different approaches, of which I found the above one among the simplest. An even simpler way is by figuring out how the original magic square was created in the first place.

I started by placing 10 numbers outside the grid as shown. In each cell inside the grid, I entered the sum of the numbers in the row and column intersecting at that cell. For example, in row 3, column 4, I entered 13 + 8 = 21.

Since the procedure effectively makes you select one number from each column and one number from each row, their total will be the sum of the 10 numbers outside the grid, which is 100.

I came across this curiosity in Mathematical Puzzles and Diversions by Martin Gardner, who uses an example with 59 as the total.

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