What Are The Math Problems in Bitcoin Mining?

Kevin Groves
Posted by: Kevin GrovesUpdated Jul 28th, 2023

Key Takeaways:

  • Mining Bitcoin is vital to securing the blockchain protocol and creating new coins.
  • Miners are tasked with validating new transactions by solving cryptographic puzzles, also known as math problems.
  • The problem all miners work on solving is called the hash.
  • Successful miners are rewarded with a fixed amount of BTC and transaction fees.

The Bitcoin blockchain enables peer-to-peer (P2P) transactions without the interference of intermediaries. This is enabled thanks to the process of Bitcoin mining. Miners are computer nodes that solve complex mathematical problems on the Bitcoin network in a bid to verify blockchain transactions. For their efforts, successful miners receive newly-minted Bitcoins.

The mathematical algorithms are crucial as they help maintain the integrity of the Bitcoin network and ensure fraudulent transactions are not published on the blockchain ledger. This article dives into the math problems used in Bitcoin mining. We will also explore what Bitcoin mining is and how it contributes to the network's security.

What Are The “Math Problems” In Bitcoin Mining?

Mining math problems are complex puzzles designed as safeguards against bad actors hacking the decentralized Bitcoin system. The process of solving these complex mathematical equations is called the proof-of-work (PoW) consensus algorithm. The math problems in this system are the bedrock of the Bitcoin network and ensure that the whole system is legitimate and authentic.

The PoW algorithm is a form of cryptographic proof where a party proves to others that a certain amount of their computing power or resource has been utilized towards solving the puzzle. Other blockchain nodes participating in the network can validate this expenditure with minimal effort. n return for solving mathematical problems or puzzles and adding verified transactions as blocks to the Bitcoin network, miners earn block rewards and fees for their efforts.

1. Bitcoin's Hashing Math Problem

The main math problem in Bitcoin mining is called the Hashing problem. The PoW system is powered by the SHA-256 hashing algorithm – a blockchain function used to deter unapproved access and ensure the safety of new blocks.

This hashing algorithm ensures all blocks generated are digitally signed and have a unique hash value. It is a one-way function, meaning all data inputted is converted to a 256-bit integer. For instance, a 512-bit integer, when exposed to this blockchain function, will be converted to a 256-bit, but it would be impossible to switch back to 512-bit. Given this, miners are tasked with inverting the 256-bit hash function by computing the hash value.

This Bitcoin mining process is quite arduous as miners need to hash the data and ensure the output is below a target value prescribed by the blockchain network. The more miners that join the network, the greater their hash power and the easier it is to solve these math problems. Bitcoin's mining difficulty determines how difficult and complex it is for miners to find the right hash for each block. Therefore, the Bitcoin network algorithmically adjusts its hash difficulty over time.

To solve these mathematical puzzles, Bitcoin miners require strong hardware or Bitcoin mining rigs. As there are so many potential solutions to the hashing problem, so far the fastest way to find an answer is via “brute force”. This basically means specialized mining machines spit out potential answers until the right solution is derived for the hashing problem. The mathematical puzzles are crucial to preserving the Bitcoin network’s legitimacy.

They ensure that no bad actor can gain a majority hold of the decentralized protocol. As miners are forced to try countless hash functions until they identify a value that works, gaining enough nodes to control 51% of the Bitcoin network would be extraordinarily time-consuming and expensive. This is also how the Bitcoin mining process resolves the double-spending scenario.

2. Double-spend Problem

Double-spending occurs when a currency is spent more than once. In the “real world”, this would be like using a $20-dollar bill to pay for dinner and then using that same bill to pay for drinks. In blockchain terms, it can happen when a spender modifies the blocks on the network and reclaim the spent coins for future use.

To do this, the bad actor would need to control more than 50% of the network’s computational power and create a secret block. This block would need to be mined faster than the most recent block in the actual blockchain. But due to the distributed nature of Bitcoin mining (ownership history, consensus from all nodes, and a timestamp server), double-spending on the protocol is almost impossible.

double spend problem bitcoindouble spend problem bitcoin

3. Byzantine Fault Tolerance (BFT)

Bitcoin’s mathematical puzzles also aid in dealing with the Byzantine generals' challenge. This game theory concept seeks to address the issue around distributed networks. Since there is no central authority and participants are often anonymous, how can the truth be ascertained across such a vast group of individuals?

To address this potential issue, the concept of Byzantine Fault Tolerance (BFT for short) was introduced. This ideology posits that even if one or more computer nodes in a network are corrupt or tampered with, the network should still be legitimate. The PoW consensus algorithm system epitomizes this concept, as even when one or more nodes are offline or compromised, the miners as a whole must agree that a transaction is valid before it can be added to the Bitcoin blockchain.

Bitcoin mining software checks historical information, hash value, and details of the previous blocks to ensure the new one follows the same format before it is accepted. If there are contradictory data or transactions, these miners have the power to reject malicious transactions, thereby safeguarding the Bitcoin network.

Why Does Bitcoin Need Complex Mathematical Equations?

There are several reasons why these complex mathematical equations are needed to safeguard the Bitcoin blockchain.

  • Double-spending. The first reason is to prevent the ‘double-spending’ issue which is a risk to any blockchain network as it allows users to spend the native token twice by reclaiming the coins. If this occurs, the blockchain’s integrity is significantly damaged. The Bitcoin network mitigates this risk through its decentralized miners. If a transaction looks suspicious or does not follow the expected format, it won’t achieve consensus and is ultimately rejected. Additionally, the chances of a modified or secret block validating a transaction before the Bitcoin network is far-fetched, given that the network updates in real-time as nodes work in unison.
  • 51% Attacks. A subset of the double-spending problem, the 51% attack, is another key reason why complex mathematical equations are necessary for the continued integrity of the Bitcoin network. A 51% attack occurs when a single party controls over 50% of the network computing power. With so much power vested in a central authority, transaction information can easily be altered, disrupting the consensus algorithm and how block rewards are awarded. With the Bitcoin blockchain, this is hard to achieve due to the distributed mining node system and the hash rate difficulty.
  • Distributed Denial-of-Service Attack. A distributed denial-of-service (DDoS) attack is an attempt to render a network inaccessible by flooding it with multiple requests to overwhelm it. These multiple network requests often stem from a single party. In the case of a blockchain service, the attacker can be quite difficult to track down due to the network’s distributed nature. To address this issue, a transaction fee is imposed – so a DDoS attack would become quite expensive given the number of transactions required to down the Bitcoin network.
  • Relay Attack. Finally, a relay attack is another blockchain-specific challenge that requires complex mathematical equations. This occurs when a bad actor intercepts and relays verified data across a network. The network security would naturally accept the data since it comes from a verified source. However, this could see the malicious actor gaining access to a secured network after bypassing the network’s firewalls. Cybercriminals could use this to steal tokens or undermine the network integrity outright. That said, these attacks are quite easy to counter thanks to the Bitcoin network using timestamps when it transfers data.
math problem to solve bitcoin trust issuemath problem to solve bitcoin trust issue

Do You Need A Supercomputer To Solve Bitcoin's Math Problems?

A popular misconception is that Bitcoin math problems can only be solved by supercomputers. While the difficulty to mine Bitcoin has increased dramatically since its inception in 2009, this is still not quite true. To validate transactions and mine Bitcoin, Bitcoin miners use specialized hardware mining rigs, not supercomputers.

The most popular hardware for mining is Application Specific Integrated Circuit (ASIC) rigs. These computerized devices use integrated chips to mine specific cryptocurrencies like Bitcoin. ASIC miners are inherently faster and more efficient than the earlier generations of mining outlets due to their specialized nature.

Given the deflationary tendency of Bitcoin, the asset value has increased over the years, attracting more miners to join the network. As such, ASIC miners have become the most powerful and efficient means to crack the hashing problem and enable cryptographic trust throughout the network.

Prior to ASIC miners, Bitcoin mining was mainly done using personal computers via the central processing unit (CPU) and graphics processing unit (GPU). While both were effective tools for mining, the growth of the Bitcoin network led to more miners coming on board and the hash rate difficulty increasing concurrently.

Related: How to build your own ASIC miner in 6 steps.

What Are The Rewards for Solving Bitcoin Math Problems?

The reward for solving these mathematical puzzles is the creation of new coins through mining and transaction fees for the network’s miners. These new coins are released as a block reward for validating network transactions. But why do miners get rewarded for verifying transactions? There are a few reasons.

For one, miners help secure the network from potentially debilitating attacks. Additionally, mining is a key element of Bitcoin’s tokenomics and is how the blockchain regulates supply and demand. It's the only appropriate means to release new Bitcoins into the market, making the practice even more important for the continued sustainability of the digital asset. This ensures liquidity is readily available for investors looking to trade the virtual asset.

The first four years following Bitcoin’s emergence onto the world stage saw miners earn 50 BTC while conveniently using their CPUs and GPUs. However, Bitcoin's halving event, which reduces block rewards in half every four years, has reduced the incentives over time.

In 2023, the reward stands at 6.25 BTC. The next Bitcoin halving event is scheduled for 2024, which will slash the block reward by half again. While this incentive rate might seem smaller relative to the first payout of 50 BTC in 2012, Bitcoin's value has more than quadrupled from its lowly value of $0.10 in 2009 when it launched. A Glassnode report noted that miners have earned over $50.2 billion in block rewards and transaction fees since Bitcoin’s launch, pointing to a booming practice.

How Does Bitcoin Work With Math Problems?

Bitcoin mining is a crucial part of how the Bitcoin network operates. Miners that solve complex mathematical problems are responsible for validating Bitcoin transactions and adding relevant data to the ledger. In exchange for solving these complex puzzles, miners are rewarded with newly minted Bitcoins – adding to Bitcoin’s circulating supply.

A node is a computer that runs the Bitcoin software – essentially a copy of the blockchain. They send, receive, validate and broadcast transactions with other blockchain nodes and miners to maintain the integrity of the blockchain. Since the Bitcoin protocol is decentralized, the mining process ensures that these independent nodes vet all pending transactions.

Miners select the transactions they intend to verify, often based on the Bitcoin transaction fees attached. The higher the transaction fees, the higher the chances of the transaction being verified on time. The mining software will often choose the most lucrative transaction automatically, so the miner doesn’t have to sit at their computer 24/7. The miner then competes with other miners to be the first to successfully verify the transaction.

Once a miner succeeds, it broadcasts that transaction to the rest of the network for further review. If it passes through the broader network review, it is added as a transaction in a new block. An average Bitcoin block contains about 3,000 transactions and takes around 11 minutes to send. Once the transaction block size is filled, it is sent to the blockchain and attached to the most recent block, forming a chain.

Frequently Asked Questions

Bitcoin miners solve complex cryptographic hash puzzles through a “brute force” method. This means mining rigs must output thousands (or millions) of answers until they find the right one to solve the math problem in order to validate new transactions on the blockchain.

Miners are tasked with solving a hashing problem, which them miners to find the correct inverse value for a designated hash. Once a miner successfully solves this equation, they get a block reward (newly-minted coins), and the transaction fee is shared as commissions with others.


This guide has addressed the math problems in Bitcoin mining. We have explored the key reasons these complex cryptographic puzzles are necessary and how they can impact the security and integrity of the network.

However, Bitcoin mining requires significant power and capital investment to set up. As a result, it may be a challenging pursuit for casual crypto enthusiasts without deep pockets. It's worth noting that anyone interested in joining the decentralized ledger network can still earn block rewards since the last Bitcoin will be mined in approximately 2140.

Ben KnightBen KnightBen Knight

Ben Knight is a freelance writer with a passion for music and finance from Melbourne, Australia. He completed his Master's in Writing, Editing, and Publishing in 2019, following a Bachelor's degree in Creative Writing. Ben has been writing in the finance and cryptocurrency industry for more than three years.

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