We present three very simple variants of the classic Heads or Tails game using chips, each of which contributes to our understanding of the Bitcoin protocol. The first variant addresses the issue of temporary Bitcoin forks, which occur when two miners discover blocks simultaneously. We determine the threshold at which an honest but temporarily ``Byzantine'' miner persists in mining on their fork to save his orphaned blocks. The second variant of Heads or Tails game is biased in favor of the player and helps to explain why the difficulty adjustment formula is vulnerable to attacks of Nakamoto's consensus. We derive directly and in a simple way, without relying on a Markov decision solver as was the case until now, the threshold beyond which a miner without connectivity finds it advantageous to adopt a deviant mining strategy on Bitcoin. The third variant of Heads or Tails game is unbiased and demonstrates that this issue in the Difficulty Adjustment formula can be fully rectified. Our results are in agreement with the existing literature that we clarify both qualitatively and quantitatively using very simple models and scripts that are easy to implement.

翻译：我们提出三种基于筹码的经典“猜硬币”游戏简便变体，每种变体均有助于理解比特币协议。第一种变体探讨比特币临时分叉问题——即两名矿工同时发现区块的情形。我们确定了诚实但暂时“拜占庭”的矿工为保留其孤块而持续挖掘分叉的阈值。第二种“猜硬币”游戏变体对玩家存在偏向，有助于阐释为何难度调整公式易受中本聪共识攻击。我们通过直接且简洁的方式（无需依赖当前研究常用的马尔可夫决策求解器）推导出：无连接性的矿工在比特币网络中采用异常挖矿策略的有利阈值。第三种“猜硬币”游戏变体无偏向性，证明难度调整公式的该问题可被完全修正。本研究结果与现有文献吻合，我们通过易于实现的简洁模型与脚本，从定性与定量角度对既有研究进行了阐明。