Hong Life CalculatorsUnderstanding the proof-of-work difficulty calculator — concept and principle
In proof-of-work blockchains such as Bitcoin, miners repeatedly hash a block header until the result falls below a target. The network difficulty says how hard that is. This calculator turns difficulty and your hash rate into the expected time for you to find a block alone, and estimates the total network hash rate that the difficulty implies. It shows why solo mining is a lottery.
How it is calculated
- Expected hashes per block ≈ difficulty × 2³² (2³² = 4,294,967,296). At difficulty 1, a miner needs about 2³² hashes on average.
- Expected time = difficulty × 2³² ÷ your hash rate, with your hash rate converted from TH/s to hashes per second (× 10¹²).
- Implied network hash rate = difficulty × 2³² ÷ 600 s, shown in EH/s (10¹⁸ hashes per second), because Bitcoin aims for one block every 10 minutes.
The principle
A good hash function behaves like a random number generator: each attempt has the same small chance of landing below the target, independent of earlier attempts. The number of tries until success therefore follows a geometric distribution whose mean is 1 ÷ (chance per try), and the difficulty is defined so that this mean equals about difficulty × 2³². Bitcoin retargets the difficulty every 2,016 blocks so that, whatever hardware joins, blocks keep arriving roughly every 10 minutes. That is why difficulty, multiplied out, reveals the network's total hash power.
Worked example
- Inputs: difficulty 1.0 × 10¹⁴, hash rate 200 TH/s = 2 × 10¹⁴ H/s.
- Expected hashes = 10¹⁴ × 4,294,967,296 ≈ 4.295 × 10²³.
- Time = 4.295 × 10²³ ÷ 2 × 10¹⁴ ≈ 2.147 × 10⁹ s.
- 2.147 × 10⁹ s ÷ 86,400 ÷ 365.25 ≈ 68 years.
- Network hash rate = 4.295 × 10²³ ÷ 600 ≈ 7.158 × 10²⁰ H/s = 715.8 EH/s.
| Your hash rate (TH/s) | Expected time to find a block |
|---|---|
| 50 | 272 years |
| 100 | 136 years |
| 150 | 90.7 years |
| 200 | 68 years |
| 250 | 54.4 years |
| 300 | 45.4 years |
| 400 | 34 years |
How to read the result
"68 years" is an average, not a promise. Block discovery is memoryless, so the chance of finding at least one block in time t is 1 − e^(−t / expected time); for one year here that is about 1.5%. Your share of the network is your hash rate divided by the network rate (here about 0.00003%), which is also your expected share of block rewards in a pool, before fees.
Common mistakes
- Mixing units: TH/s is 10¹², PH/s 10¹⁵, EH/s 10¹⁸.
- Expecting steady income from solo mining; results are highly random.
- Ignoring electricity, hardware and pool costs, and the fact that difficulty usually rises over time.
Frequently asked questions
Why is it 2³² and not exactly 2³²?
Difficulty 1 corresponds to a target of 0xFFFF × 2²⁰⁸, so the exact factor is 2⁴⁸ ÷ 65,535, which is very slightly more than 2³². The difference is negligible.
Does joining a pool increase my rewards?
Not on average. A pool smooths your income into frequent small payments; the expected total stays the same minus the pool fee.
How often does Bitcoin difficulty change?
Every 2,016 blocks, about two weeks, based on how long the previous 2,016 blocks took.
Sources: Satoshi Nakamoto, "Bitcoin: A Peer-to-Peer Electronic Cash System", 2008 · A. Narayanan et al., Bitcoin and Cryptocurrency Technologies, Princeton University Press, 2016