Choice and chance

Drop7 gives you a disc and a decision: where should it go? A move can clear a few discs straight away, or leave a useful arrangement for later. Learning to see both is the start of playing ahead.

The idea

A disc falls down the column you choose. It stops at the bottom or on another disc. Then the game checks for matches.

A numbered disc clears when its number equals the number of touching discs in its row or column, including itself. The other discs can have any number. A gap ends the count. An isolated 1 clears; a 2 beside one other disc can clear.

Clears leave gaps. The discs above fall, which can create more matches. Each new round of clears is a wave. Later waves in the same chain earn more points per disc.

This is the part you control: the column. After the move settles, the game deals another disc. That is the part you do not control: chance.

A small example

A real position

Here is the same board, with the same 1 waiting above it. Follow a branch to see what happens when you choose that column. Every branch starts from the board at the top. Watch the replays, or use “Next step” to inspect the clears one at a time.

Column 6 is tempting. The 1 and 4 clear in the first wave for 14 points. The 2 then falls beside the 3 and clears for another 39. Together, that is 53 points. It is the best move if we only count what happens now.

Column 1 earns just 7 points. The 1 clears on its own, leaving the rest of the board as it was. That leaves something to work with on the next turn.

How it works

The disc you do not know yet

In the Hardcore version of Drop7 used here, the next disc can be any number from 1 to 7. Each has the same chance: one in seven.

So imagine each possible disc in turn. For each one, find the reply that earns the most points. You get to choose that reply after you see the disc.

The two boards below start after our first move. Select a possible next disc to watch its best reply on each board. Start with the 2: the quieter first move has left a much bigger chain available.

Giving every possibility its share

A 2 would be helpful, but it might not arrive. To give all seven possibilities the same weight, add their best reply scores and divide by seven. That is the average reply.

Add the points from the first move to that average. The result is the move’s expected value over these two moves. It describes an average across the possible next discs; a single play will produce one of those outcomes.

Column 1 has an average reply worth about 69.1 points, giving it a two-move value of about 76.1. Column 6 gives about 67.6. Looking one turn further changes the choice.

The buttons show what happens when we change the question. Counting only today’s points favors column 6. Hoping for the luckiest disc favors column 4. Preparing for the least helpful disc favors column 6. Averaging all the possibilities favors columns 1, 2 and 3 equally.

This pattern has a name: expectimax. At a choice, take the best available move. At a chance event, average the possible outcomes. The name can wait; those two actions are the idea to keep.

In Drop7

Looking further ahead

We can repeat the same process for another turn. Choose a column, consider what might be dealt, choose a reply, and continue. The number of moves we look ahead is the search’s depth.

Each extra turn creates more possibilities to examine. Eventually the search has to stop. It then needs an estimate of how useful the remaining board is. That estimate is called a board evaluator.

The next lesson, evaluating a board, looks at that question. You can also try playing or see the full rules, including gray discs and the rising floor.

What it cannot do

A better average does not promise a better result on every play. You can make a sensible choice and still get an unhelpful disc.

This example compares the points from two moves on a small board. A full game also asks you to make room for new discs and survive the rows that rise from below. A move that earns more points soon can still leave a difficult position later.

Gray discs add another kind of chance. Their numbers are hidden until clears beside them open them. A player has to consider those possible reveals too. This teaching board has no gray discs, so we can focus on the next deal first.

The skill is to make a choice using what you can see, while leaving room for what you cannot know yet.

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