> For the complete documentation index, see [llms.txt](https://shai-deshe.gitbook.io/pow-book/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://shai-deshe.gitbook.io/pow-book/part-1-blockchains-and-blockdags/chapter-2-the-block-chain-paradigm/exercises.md).

# Exercises

1. Prove that a (simple undirected graph) is a DAG if and only if there are no two vertices $$A$$ and $$B$$ such that $$A$$ is reachable from $$B$$ *and* $$B$$ is reachable from $$A$$
2. Prove that a rooted DAG has only one root
3. Show that a rooted DAG is a tree if and only if for any block $$B$$ that is not the root, there is *a single* path from $$B$$ to the root
4. Is the following a good tie-breaking rule? Given a tie between $$A$$ and $$B$$, mine over the block that has an earlier timestamp.
5. Recall that a *satoshi* is *one hundred millionth* of a bitcoin. Assuming that Bitcoin stops coin emission when the reward drops below satoshi, how long will it take for all rewards to be emitted since the network started? By how much the total supply was decreased by discarding sub-satoshi block rewards?
6. \* Show that if we apply the Eyal-Sirer strategy to a miner with more than half of the hash rate we recover the standard double-spending attack.