Chess starts simple.
On the very first move, you only have 20 options.
But just a few turns later, the number explodes.
So here's the question:
How many legal move sequences are possible in the first 3 turns of chess?
That means:
- White moves
- Black moves
- White moves
- Black moves
- White moves
- Black moves
A total of 6 moves.
Step 1 - The First Turn
At the start of a chess game:
- each of the 8 pawns can move either 1 square or 2 squares
-> 16 possible moves
- each of the 2 knights can move in 2 ways
-> 4 possible moves
So White has:
👉 20 possible moves
Black then has the same:
👉 20 possible replies
So after one full turn each:
20 x 20 = 400
👉 400 possible sequences
Step 2 - The Game Opens Up
Now the board opens up.
Take a simple example:
- e4
... e5
That already opens up:
- the bishop
- the queen
- new attacks and responses
After just those two moves, both sides often have significantly more legal options.
So instead of around 20 moves, players now often have:
👉 ~25 to 35 legal moves
A practical way to model this is:
- Turn 2 average: ~25 moves
- Turn 3 average: ~30 moves
Step 3 - Extend to 3 Turns
We are counting 6 moves in total:
- Turn 1: White + Black
- Turn 2: White + Black
- Turn 3: White + Black
Using our rough logic:
- first two moves: 20 x 20
- turn 2 (White + Black): 25 x 25
- turn 3 (White + Black): 30 x 30
So a solid estimate could be:
20 x 20 x 25 x 25 x 30 x 30 = 225,000,000
👉 ~225 million move sequences
Final Estimate
If you landed somewhere around:
👉 200 to 250 million
that is excellent reasoning.
You've correctly:
- identified the small starting point (20)
- recognized that the game opens up (~30)
- applied compounding branching
What Is the Real Number?
The exact number of legal move sequences in the first 3 turns of chess is:
👉 119,060,324
This comes from chess perft counts, which fully enumerate every legal move path.
Magnitudle Score
Let's compare:
- Your estimate: 225,000,000
- Actual: 119,060,324
That means the estimate is:
👉 about 1.9x too high
That is still less than half an order of magnitude off.
👉 Magnitudle Score: 93 / 100
Why the Estimate Ends Up Slightly High
The logic is strong, but reality is messier.
Not every position has around 30 moves.
Some positions:
- are blocked
- include checks or forcing replies
- allow captures that reduce later options
In some lines, a player can have only one legal move.
For example, when in check, there are positions where only a single move gets out of check.
So while 30 is a very good average guess, the true branching factor is slightly lower overall.
Why This Works
This estimate works because it combines:
- a known exact starting point (20 legal opening moves)
- a realistic branching estimate once pieces develop
- compounding across multiple turns
Even with rough assumptions, it lands in the right magnitude range.
Final Thought
This is what makes chess so fascinating.
It starts with just 20 moves.
But after only 3 turns each:
👉 there are already over 119 million possible games
Fun Fact
These exact counts are used by chess engine developers to test move generation.
If the number is wrong:
👉 the engine is broken
Try a Question Like This Yourself
Questions like this are exactly what you'll find on Magnitudle.
Each day you get one real-world question and one chance to estimate the answer.
The closer you are to the true magnitude, the higher your score.