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Merging cards

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The numbers $1$ , $2$ , $3$ , $4$ , $5$ and $6$ are written on cards on a table. A move takes any $2$ cards with numbers $a$ and $b$ and replaces them by one card with $a + b + ab$ . After $5$ moves one card remains. What number is on it?

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Hint

Add $1$ to the new number and factor.

Solution

Note that $a + b + ab + 1 = (a + 1) (b + 1)$ .
So the product of the values $x + 1$ over all the cards never changes.
At the start this product is $2 \times 3 \times 4 \times 5 \times 6 \times 7 = 5040$ .
The last card $x$ therefore has $x + 1 = 5040$ , so $x = 5040 - 1 = 5039$ .

Answer: $5039$

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