What interviewers evaluate when asking puzzles:

1. Structured thinking
   → Can you break a complex problem into parts?

2. Communication under pressure
   → Can you think out loud clearly?

3. Comfort with ambiguity
   → How do you react when there is no obvious path?

4. Creativity
   → Can you see non-obvious approaches?

5. Intellectual curiosity
   → Do you enjoy the problem or just endure it?

What they do NOT test:
✗ Prior knowledge of the puzzle
✗ Speed of getting the answer
✗ Mathematical genius

The interviewer wants to see HOW you think,
not IF you get the right answer.

Solution:

Ask either guard: "If I asked the OTHER guard which
door leads to freedom, what would they say?"

Then choose the OPPOSITE door.

Why this works:
- If you ask the truth-teller:
  He truthfully reports the liar's answer (wrong door)
- If you ask the liar:
  He lies about the truth-teller's answer (wrong door)

Either way, you get the WRONG door pointed out.
So you pick the other one.

What the interviewer evaluates:
→ Can you use meta-reasoning (question about a question)?
→ Can you verify your logic covers both cases?

Solution:

1. Turn switch 1 ON for 10 minutes
2. Turn switch 1 OFF
3. Turn switch 2 ON
4. Enter the room

If bulb is ON → Switch 2
If bulb is OFF but WARM → Switch 1
If bulb is OFF and COLD → Switch 3

Key insight: Use HEAT as a second data point.
Most people only think about on/off (1 bit).
Temperature gives you a second bit of information.

What the interviewer evaluates:
→ Can you think beyond the obvious binary state?
→ Do you consider physical properties, not just logic?

Solution: Binary encoding

Number each bottle 1-1000 in binary (10 bits).
Each prisoner represents one bit position.

Bottle 1   = 0000000001 → only prisoner 1 drinks
Bottle 2   = 0000000010 → only prisoner 2 drinks
Bottle 3   = 0000000011 → prisoners 1 and 2 drink
...
Bottle 1000 = 1111101000 → prisoners 4,6,7,8,9,10

After 24 hours, the pattern of dead prisoners
gives you the binary number of the poisoned bottle.

Example: if prisoners 3 and 7 die → binary 0001000100
= bottle 68

Why 10 prisoners is enough: 2^10 = 1024 > 1000

What the interviewer evaluates:
→ Do you know binary encoding?
→ Can you map abstract CS concepts to real problems?

Solution:

At time 0:
- Light Rope 1 from BOTH ends
- Light Rope 2 from ONE end

At time 30 minutes:
- Rope 1 is completely burned (60/2 = 30 min)
- Rope 2 has 30 minutes of burn left

When Rope 1 finishes (30 min mark):
- Light Rope 2 from the OTHER end too

Rope 2 now burns from both ends.
It had 30 minutes left → now takes 15 minutes.

Total: 30 + 15 = 45 minutes ✓

Key insight: Lighting from both ends halves the time,
regardless of uneven burning.

What the interviewer evaluates:
→ Can you combine two simple ideas creatively?
→ Do you handle the "uneven burning" constraint?

Solution (7 trips):

Trip 1: Farmer takes CHICKEN across →
        (fox + grain safe together)
Trip 2: Farmer returns alone ←
Trip 3: Farmer takes FOX across →
Trip 4: Farmer brings CHICKEN back ←
        (key insight: bring something BACK)
Trip 5: Farmer takes GRAIN across →
        (fox + grain safe, chicken on other side)
Trip 6: Farmer returns alone ←
Trip 7: Farmer takes CHICKEN across →

All safe on the other side ✓

The trick: most people do not consider bringing
an item BACK. The chicken acts as a "swap."

What the interviewer evaluates:
→ Can you think about state management?
→ Do you consider non-obvious moves (going backward)?

Structured approach:

Step 1: Estimate room dimensions
  → Typical interview room: 4m x 5m x 3m
  → Volume = 60 cubic meters = 60,000,000 cm³

Step 2: Estimate tennis ball size
  → Diameter ≈ 6.7 cm, radius ≈ 3.35 cm
  → Volume of one ball ≈ (4/3)π(3.35)³ ≈ 157 cm³

Step 3: Account for packing efficiency
  → Random packing ≈ 64% efficiency
  → Usable volume = 60,000,000 × 0.64 = 38,400,000 cm³

Step 4: Calculate
  → 38,400,000 / 157 ≈ 244,586 balls
  → Round to ~250,000 tennis balls

State your assumptions clearly.
The exact number does not matter — the structure does.

Structured approach:

Step 1: India's population → ~1.4 billion
Step 2: Vehicles per person
  → ~1 vehicle per 5 people = 280 million vehicles
  → Mix of 2-wheelers (70%), cars (25%), commercial (5%)

Step 3: Fuel consumption per vehicle per month
  → Average: ~50-80 liters/month across types

Step 4: How much can one pump sell?
  → Average pump sells ~100,000-150,000 liters/month

Step 5: Total fuel demand / per-pump capacity
  → ~280M vehicles × 60L avg = 16.8 billion liters/month
  → 16.8B / 125,000 = ~134,000 pumps

Actual number: ~84,000 (as of 2024)
Being within 2x is considered a good Fermi estimate.

Key: Show your reasoning chain, not just a guess.

Structured approach:

Step 1: WhatsApp users in India → ~500 million
Step 2: Daily active users → ~70% = 350 million

Step 3: Segment users by activity:
  → Heavy users (20%): 100+ messages/day
  → Medium users (40%): 30-50 messages/day
  → Light users (40%): 5-10 messages/day

Step 4: Calculate weighted average
  → 70M × 100 = 7 billion
  → 140M × 40 = 5.6 billion
  → 140M × 7 = 980 million
  → Total ≈ 13.6 billion messages/day

Step 5: Add group messages (multiplier effect)
  → Groups amplify by ~1.5x
  → ~20 billion messages/day

This is an India-specific question — show you
understand Indian digital behavior.

Structured approach:

Step 1: Think about components
  → Fuselage (aluminum/composite body)
  → Wings and engines
  → Fuel
  → Passengers and cargo

Step 2: Estimate each
  → Empty aircraft structure: think of it as a
    large aluminum tube, 70m long, 6m diameter
  → Engines: 4 engines, each ~5,000 kg = 20,000 kg
  → Fuel capacity: ~200,000 liters of jet fuel
    (density ~0.8 kg/L) = 160,000 kg
  → Passengers: ~400 people × 80 kg = 32,000 kg
  → Cargo and equipment: ~20,000 kg

Step 3: Empty weight estimate
  → Structure + systems ≈ 180,000 kg

Step 4: Maximum takeoff weight
  → 180,000 + 160,000 + 32,000 + 20,000
  → ≈ 392,000 kg ≈ 400,000 kg (400 tonnes)

Actual: ~412,000 kg (max takeoff weight)
Excellent estimate!

Structured approach (classic Fermi question):

Step 1: Mumbai population → ~21 million
Step 2: How many pianos in Mumbai?
  → Pianos are rare in India compared to the West
  → Hotels with pianos: ~200 hotels × 1 = 200
  → Music schools/churches: ~300
  → Wealthy homes: ~5,000 (very rough)
  → Recording studios/venues: ~100
  → Total: ~5,600 pianos

Step 3: How often is a piano tuned?
  → Professional pianos: 2-4 times/year
  → Home pianos: 1-2 times/year
  → Average: 2 times/year
  → Total tunings: 5,600 × 2 = 11,200/year

Step 4: How many tunings can one tuner do?
  → 4 tunings/day × 250 working days = 1,000/year

Step 5: Tuners needed
  → 11,200 / 1,000 ≈ 11 piano tuners

This is a tiny number — which makes sense for
Mumbai where pianos are not common instruments.

Solution: YES, always switch.

Probability if you STAY:  1/3 (your initial pick)
Probability if you SWITCH: 2/3

Why? Your initial pick has 1/3 chance of being right.
That means there is a 2/3 chance the car is behind
one of the other two doors. When the host opens a
goat door, all that 2/3 probability concentrates
on the remaining door.

Think of it with 100 doors:
- You pick door 1 (1% chance)
- Host opens 98 doors showing goats
- Would you switch to the one remaining door?
  Of course! It has 99% probability.

Common mistake: thinking it is 50/50 after the reveal.
The host's action gives you NEW information.

Solution: Only 23 people.

This is counterintuitive because we think about
the chance of someone sharing OUR birthday (1/365).
But the question is about ANY two people matching.

With 23 people, there are C(23,2) = 253 pairs.
Each pair has a 1/365 chance of matching.
253 pairs × (1/365) ≈ 69% (rough estimate)

Exact calculation (complement method):
P(no match) = 365/365 × 364/365 × 363/365 × ...
For 23 people: P(no match) ≈ 0.493
P(at least one match) = 1 - 0.493 = 0.507 > 50% ✓

At 50 people: probability rises to 97%
At 70 people: 99.9%

Key insight: combinatorial explosion of pairs
grows much faster than linear addition of people.

Solution: 6 flips (expected value)

Let E = expected flips to get HH

State analysis:
- Start state (S): need 2 heads
- State H: got 1 head, need 1 more
- State HH: done!

From S:
  Flip H (prob 1/2) → go to state H
  Flip T (prob 1/2) → stay in S
  E_S = 1 + (1/2)E_H + (1/2)E_S

From H:
  Flip H (prob 1/2) → done! (0 more needed)
  Flip T (prob 1/2) → back to S
  E_H = 1 + (1/2)(0) + (1/2)E_S

Solving:
  E_H = 1 + (1/2)E_S
  E_S = 1 + (1/2)(1 + (1/2)E_S) + (1/2)E_S
  E_S = 1 + 1/2 + (1/4)E_S + (1/2)E_S
  E_S = 3/2 + (3/4)E_S
  (1/4)E_S = 3/2
  E_S = 6

Expected flips = 6

Solution: Follow the cycle strategy (~31% success)

Strategy:
1. Prisoner #k starts by opening box #k
2. If box #k contains number m, open box #m next
3. Follow this chain until you find your number
4. Stop after 50 boxes

Why this works:
The random arrangement creates permutation cycles.
If no cycle is longer than 50, ALL prisoners succeed.

P(longest cycle ≤ 50) ≈ 1 - ln(2) ≈ 0.3069 ≈ 31%

Random guessing: (1/2)^100 ≈ 0.0000...0008
Cycle strategy: ~31%

This is astronomically better than random!

Key insight: by following cycles, prisoners' fates
become correlated (all succeed or all fail together)
instead of independent.

Solution:

Let E = expected value with optimal strategy.

If you roll and get value v:
  Keep if v ≥ E (the value beats re-rolling)
  Re-roll if v < E (pay Rs 1 for another chance)

Start with guess: E ≈ 4
  Keep: 4, 5, 6 (probability 3/6 = 1/2)
  Re-roll: 1, 2, 3 (probability 1/2)

E = (1/2) × average(4,5,6) + (1/2) × (E - 1)
E = (1/2)(5) + (1/2)(E - 1)
E = 2.5 + E/2 - 0.5
E = 2 + E/2
E/2 = 2
E = 4

Verify: keep 4,5,6 → average kept = 5
E = (1/2)(5) + (1/2)(4-1) = 2.5 + 1.5 = 4 ✓

Optimal strategy: keep 4, 5, or 6. Re-roll 1, 2, 3.
Expected value: Rs 4 per game.