Running bitcoin 
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1. 21 Million Supply Limit
Bitcoin’s total supply is capped at 21 million BTC. This comes from the block reward halving schedule:
Initial block reward: 50 BTC
Halving every 210,000 blocks (~4 years)
Total BTC = sum of geometric series:
total_btc = 0
block_reward = 50
halving_interval = 210000
while block_reward > 0:
total_btc += block_reward * halving_interval
block_reward /= 2
print(total_btc) # Should output 21000000
2. Block Size Limit
Originally, Bitcoin had a 1 MB block size limit (set by Satoshi). Transactions in a block cannot exceed this limit.
MAX_BLOCK_SIZE = 1_000_000 # bytes
def is_block_valid(block):
return block.size <= MAX_BLOCK_SIZE
3. Difficulty Adjustment
Bitcoin adjusts the mining difficulty every 2016 blocks (~2 weeks) to keep the block interval ~10 minutes.
Formula:
\text{New Difficulty} = \text{Old Difficulty} \times \frac{\text{Actual Time of Last 2016 Blocks}}{20160 \text{ minutes}}
If blocks came too fast → difficulty ↑
If blocks came too slow → difficulty ↓
def adjust_difficulty(prev_difficulty, actual_time_seconds):
target_time = 2016 * 10 * 60 # 2016 blocks * 10 minutes per block
new_difficulty = prev_difficulty * (actual_time_seconds / target_time)
return new_difficulty
4. Block Interval (~10 minutes)
Bitcoin targets a 10-minute interval between blocks.
This is enforced indirectly via mining difficulty: miners must solve Proof-of-Work (PoW) hashes under the target.
Formula for target hash:
\text{hash} \leq \text{target}
import hashlib
def mine_block(data, target):
nonce = 0
while True:
hash_result = hashlib.sha256((data + str(nonce)).encode()).hexdigest()
if int(hash_result, 16) <= target:
return nonce, hash_result
nonce += 1
5. Consensus Rules for Valid Transactions
A transaction is valid if it meets Bitcoin’s rules:
1. Inputs reference unspent outputs (UTXOs)
2. Input amounts ≥ output amounts
3. Signatures are valid
4. ScriptPubKey conditions satisfied
5. No double-spending
def is_tx_valid(tx, utxos):
input_sum = 0
output_sum = 0
for inp in tx.inputs:
if inp.utxo not in utxos:
return False # input does not exist
if not verify_signature(inp.signature, inp.utxo.pubkey):
return False # invalid signature
input_sum += inp.utxo.amount
for out in tx.outputs:
output_sum += out.amount
if input_sum < output_sum:
return False # trying to create coins out of nothing
return True
If consensus fails, Bitcoin loses integrity.
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