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What's the best way to estimate or determine the approximate number of systems doing mining and maintaining the integrity of the ledger (there's https://ethstats.net/, but that doesn't show all miners, of course). Also curious about ways to know the number of smart contracts deployed on the mainnet.

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The best way to estimate the approximate number of systems doing mining is to use the hashrate or difficulty.

From https://etherscan.io/charts/hashrate, the current hashrate is 5,479 GH/s, or 5,479,000 MH/s. For comparison, my 4 x R9 390x produces 125 MH/s. So roughly 43,832 x my mining rig.

The current number of ethereum nodes maintaining the blockchain data is 6,228 - from http://www.ethernodes.org/network/1.

The number of smart contracts currently deployed on mainnet is 124,349. You can find the statistics from https://etherscan.io/accounts/c . An unknown proportion of these smart contracts would be the Wallet contracts.

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