Controlling 1000 Bulbs with Just 10 Switches

 A man has 1000 bulbs in his house but only 10 switches.

He wants to arrange the bulbs so that by flipping the switches in different ON/OFF combinations, he can light up any exact number of bulbs from 1 to 1000. How should he distribute the bulbs among the 10 switches?

🧮 Step‑by‑Step Solution

Key Idea: Binary Representation

  • Each switch represents a binary digit (bit).

  • With 10 switches, there are 210=1024 possible ON/OFF combinations.

  • That’s enough to represent all numbers from 0 to 1023.

  • Since we only need 1 to 1000, this works perfectly.

Distribution of Bulbs

Assign bulbs to switches in powers of 2:

  • Switch 1 → controls 1 bulb

  • Switch 2 → controls 2 bulbs

  • Switch 3 → controls 4 bulbs

  • Switch 4 → controls 8 bulbs

  • Switch 5 → controls 16 bulbs

  • Switch 6 → controls 32 bulbs

  • Switch 7 → controls 64 bulbs

  • Switch 8 → controls 128 bulbs

  • Switch 9 → controls 256 bulbs

  • Switch 10 → controls 489 bulbs (to make the total exactly 1000)

Verification

  • To light 1 bulb → turn ON switch 1.

  • To light 37 bulbs → 32 + 4 + 1 (switches 6, 3, 1).

  • To light 999 bulbs → 489 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2.

  • To light 1000 bulbs → all switches ON.

Thus, any number between 1 and 1000 can be achieved.

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