three guards were protecting an orchard. A third met the guards, one after the other, To each guard he gave half of the apples he has at the time and two extra . Eventually he escaped with just one apple. How many apples did the thief take? Fellow up questions for extension. What if he had 2 left? 3 left? 4 left?

The thief took 36 apples.

Step-by-step explanation:

This problem can be solved from end to beginning.

The thief ends up with a single apple. This happed after he gave half of the apples he had and two extra apples.

Thus, adding two apples 1+2=3 and doubling the result, the thief had 3*2=6 apples when he met the last guard.

Before that, he gave 2 apples to the second guard, thus he had 6+2=8 apples. Doubling it 8*2=16 is the number of apples he had when met the second guard.

Doing this again, 16+2=18. 18*2=36.

The thief took 36 apples.

Gave 18+2=20 to the first guard. He had 36-20=16 now.

Gave 8+2=10 to the second guard. He had 16-10=6 now.

Gave 3+2=5 to the third guard and got away with just one apple