PROOF OF WORK

There are some ways to tie your coin supply to a mathematical formula. One of the simplest ways would be to make it a "merged mining" with Ether, meaning that anyone who finds a block on Ethereum would also get a reward from your coin, given that anyone calls the reward function on that block. You can do it using the special keyword coinbase that refers to the miner who finds the block.

    function giveBlockReward() {
        balanceOf[block.coinbase] += 1;
    }

It's also possible to add a mathematical formula, so that anyone who can do math can win a reward. On this next example you have to calculate the cubic root of the current challenge gets a point and the right to set the next challenge:

    uint currentChallenge = 1; // Can you figure out the cubic root of this number?

    function rewardMathGeniuses(uint answerToCurrentReward, uint nextChallenge) {
        require(answerToCurrentReward**3 == currentChallenge); // If answer is wrong do not continue
        balanceOf[msg.sender] += 1;         // Reward the player
        currentChallenge = nextChallenge;   // Set the next challenge
    }