A thief steals half of the total number of loaves of bread plus half a loaf from a bakery. A second thief steals half of the remaining loaves plus half a loaf, and this process continues similarly for a total of five thieves. After the fifth thief, no loaves are left in the bakery. How many loaves were there in the beginning?

Introduction / Context:
This is a classic reverse working puzzle involving fractions and repeated actions. Each thief takes away a portion of the remaining loaves plus an extra half loaf, and after several such steps, nothing remains. The question checks your ability to work backwards logically through multiple stages.

Given Data / Assumptions:

• There are initially N loaves in the bakery.
• First thief takes half of N plus 0.5 loaf.
• Each subsequent thief takes half of the current number of loaves plus 0.5 loaf.
• After the fifth thief, 0 loaves remain.
• All loaves are assumed to be cuttable so that halves make sense.

Concept / Approach:
Instead of moving forward, we work backwards from the final state. If after a thief acts there are R loaves remaining, and each thief takes half of what was there before plus 0.5, we can express the loaves just before that thief came as B = 2(R + 0.5). Starting from R = 0 after the fifth thief, we repeatedly compute the previous amounts until we reach the original total.

Step-by-Step Solution:
Step 1: After the fifth thief, let the remaining loaves be R5 = 0.Step 2: Just before the fifth thief came, let that amount be B5. We know R5 = B5/2 - 0.5. Setting R5 = 0 gives 0 = B5/2 - 0.5 ⇒ B5/2 = 0.5 ⇒ B5 = 1.Step 3: After the fourth thief leaves, R4 = B5 = 1. For the fourth thief, R4 = B4/2 - 0.5. So 1 = B4/2 - 0.5 ⇒ B4/2 = 1.5 ⇒ B4 = 3.Step 4: After the third thief, R3 = B4 = 3. Then 3 = B3/2 - 0.5 ⇒ B3/2 = 3.5 ⇒ B3 = 7.Step 5: After the second thief, R2 = B3 = 7. Then 7 = B2/2 - 0.5 ⇒ B2/2 = 7.5 ⇒ B2 = 15.Step 6: After the first thief, R1 = B2 = 15. Then 15 = B1/2 - 0.5 ⇒ B1/2 = 15.5 ⇒ B1 = 31.Step 7: Therefore, the initial number of loaves N = 31.

Verification / Alternative check:
Start with 31 loaves. First thief takes 31/2 + 0.5 = 15.5 + 0.5 = 16 loaves, leaving 15. Second thief takes 15/2 + 0.5 = 7.5 + 0.5 = 8 loaves, leaving 7.Third thief takes 7/2 + 0.5 = 3.5 + 0.5 = 4 loaves, leaving 3. Fourth thief takes 3/2 + 0.5 = 1.5 + 0.5 = 2 loaves, leaving 1. Fifth thief takes 1/2 + 0.5 = 0.5 + 0.5 = 1 loaf, leaving 0. The scenario matches exactly.

Why Other Options Are Wrong:
21, 11 and 17 do not lead to zero loaves after five thieves if you apply the same process; some loaves are left or negative numbers appear, which is impossible.

Common Pitfalls:
Trying to work forward from an unknown starting value is very difficult.Some learners misapply the formula for how many loaves remain after each theft.Working backwards from the known end state is much simpler and avoids these issues.

Final Answer:
The bakery originally had 31 loaves.