Guess a Number — Thirds vs Pure Binary Strategy

This project simulates two strategies for the classic “guess a number” game:

1. Thirds → Binary Strategy

   • Split the range into 1/3 and 2/3.
   • Ask if the number is in the first third (<= N/3).
   • Only if not, ask about the second threshold (<= 2N/3).
   • Then apply binary search within the chosen third.

2. Pure Binary Strategy

   • Always split the range in half.
   • Continue dichotomically until the number is found.

We ran 1,000,000 simulations for each range and strategy:

• 1..100
• 1..1,000
• 1..10,000
• 1..100,000
• 1..1,000,000
• 1..10,000,000
• 1..100,000,000
• 1..1,000,000,000

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Results Summary

Range	Strategy	Mean	Std	Min	P50	P90	P99	Max
1..100	Thirds→Binary	6.75	0.55	6	7	7	8	8
	Pure Binary	6.72	0.45	6	7	7	7	7
1..1,000	Thirds→Binary	10.13	0.69	9	10	11	11	11
	Pure Binary	9.98	0.15	9	10	10	10	10
1..10,000	Thirds→Binary	13.44	0.63	12	14	14	14	14
	Pure Binary	13.36	0.48	13	13	14	14	14
1..100,000	Thirds→Binary	16.70	0.50	16	17	17	18	18
	Pure Binary	16.69	0.46	16	17	17	17	17
1..1,000,000	Thirds→Binary	20.09	0.68	19	20	21	21	21
	Pure Binary	19.95	0.22	19	20	20	20	20
1..10,000,000	Thirds→Binary	23.41	0.64	22	23	24	24	24
	Pure Binary	23.32	0.47	23	23	24	24	24
1..100,000,000	Thirds→Binary	26.66	0.48	25	27	27	27	27
	Pure Binary	26.66	0.47	26	27	27	27	27
1..1,000,000,000	Thirds→Binary	30.06	0.68	29	30	31	31	31
	Pure Binary	29.93	0.26	29	30	30	30	30

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Full Decile Tables (P10…P100)

Range 1..100

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	6	6	6	7	7	7	7	7	7	8
Pure Binary	6	6	7	7	7	7	7	7	7	7

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Range 1..1,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	9	10	10	10	10	10	11	11	11	11
Pure Binary	10	10	10	10	10	10	10	10	10	10

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Range 1..10,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	13	13	13	13	14	14	14	14	14	14
Pure Binary	13	13	13	13	13	14	14	14	14	14

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Range 1..100,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	16	16	16	17	17	17	17	17	17	18
Pure Binary	16	16	16	17	17	17	17	17	17	17

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Range 1..1,000,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	19	20	20	20	20	20	20	21	21	21
Pure Binary	20	20	20	20	20	20	20	20	20	20

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Range 1..10,000,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	23	23	23	23	23	23	23	24	24	24
Pure Binary	23	23	23	23	23	23	23	23	24	24

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Range 1..100,000,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	26	26	26	27	27	27	27	27	27	27
Pure Binary	26	26	26	27	27	27	27	27	27	27

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Range 1..1,000,000,000

Strategy	P10	P20	P30	P40	P50	P60	P70	P80	P90	P100
Thirds→Binary	29	30	30	30	30	30	30	30	31	31
Pure Binary	30	30	30	30	30	30	30	30	30	30

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Interpretation

• Pure Binary

  • Never exceeds ⌈log₂(N)⌉.
  • More consistent across all ranges.

• Thirds → Binary

  • Gives a 1-step advantage at P10 in several ranges (100, 1k, 1M, 1B).
  • But suffers a 1-step penalty in upper deciles (P90–P100).

👉 The pattern repeats as ranges grow:

• Advantage at ~P10.
• Disadvantage at ~P90+.
• On average, Pure Binary is better.

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How to Run

python guess_game_large.py