### Bejeweled Blitz Strategy as Math

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# Bejeweled Blitz Strategy as Math

Xah Lee, 2011-09-22

Been playing Bejeweled for like 12 hours yesterday and today.

I don't know how people do it, but i just couldn't figure it out. Normally, my score is under 50k. (currently best score of 155k, but that's once in a hundred games)

i see my friends having score 170k and 320k, and i think that's not uncommon. Looking at youtube, there are people with score of 878k!

On the other hand, been playing Diamond Dash for the past week. That one i easily mastered. Currently got top score at 358k and is the weekly champion.

Diamond Dash is fun but does not have much strategy. The key to high score is fast reflex. That's pretty much all.

Bejeweled is a bit complex. As far as i know currently, half of it is speed, but the other half is making optimal moves. (you need to keep getting the multiplier gem, and for that to happen, you need to get power jewels, and for that to happen you need to align 4 or more jewels in one shot)

I've read some tips online. The key to score high has 2 aspects: repeatedly getting the score multiplier gem, and make each move under about every 1 or 2 seconds for the speed bonus.

This seem a incredible task. I'm a thinker type of game player. If there's no time limit, i could spend several minutes per move, analyzing the optimal move. (the whole game is set for 1 minute playtime) I find it impossible to find the optimal move at the required speed.

I am guessing, that bejewel experts are not persons with machine-like lighting speed analytic mind. But rather, the mind naturally picks up patterns for optimal move, after repeated exposure. (e.g. thousands of games) I think, this is similar to the board game wei-chi (surround chess).

Am writing this because am pissed i wasn't able to raise my playing skill even after 2 days continuous play.

Here's some tips i've gathered.

## Basic Rules

I assume you already played many games.

It is strange that the bejeweled official site does not seem to have a page that tells you the exact rules and scoring system of the game. Here's some facts i gathered about the game.

There are 7 types of jewels: Blue, Green, Orange, Pink, Red, White, Yellow.

The grid is 8×8 = 64.

the above numbers are good facts to start to calculate probabilities.

### Special Gems

There are 3 special Gems: Flame Gem, Star Gem, Hypercube.

#### Flame Gem (bomb)

Flame gem explodes like a bomb.

It removes all neighboring gems. (9 of them, counting itself. But if the flame gem is on edge or corner, then less).

When you match 4 gems in a row or column, you get a flame gem.

#### Star Gem (cross-zapper)

This one is like a electric bolt gun.

It removes all gems in the same row and column, like a greek cross ✚. (15 total, counting itself)

To get a star gem, you have to match 3 vertical gems with 3 horizontal gems, in a L shape or T shape. (or it can happen in a greek cross ✚ shape during cascade.)

#### Hypercube (lightening)

Most powerful. Animated like a lightening storm.

It removes all gems on board of the same color. (the color that you swapped this gem with).

Match 5 gems in a straight line to get a hypercube.

### Multiplier Gems

Multiplier gems multiply your score. First multiplier gem is 2x, then 3x, 4x, …. For example, if you got a 2x multiplier, then every point is doubled after that.

You get a multiplier gem whenever you clear 12 or more gems in one move. However, it seems no more than 1 per 5 second. For example, if you just got a multiplier 4 seconds ago, then you made a another 12-kill in one shot, but there won't be a new multiplier.

## Key to High Score

The key to get high score is to get multiplier gems (and speed). And to get that, you practically need to get those power gems and hypercube every few moves.

Second most important is speed. When you make moves fast enough, there's speed bonus. (detail later) Basically, you'll have to spend no more than 2 second per move.

Remember, the game lasts only 60 seconds.

## Practice Tips

… working on writing this section

Currently, i see 2 aspects. One is practicing speed. You need to make a move every one or two seconds.

But then how do you know which moves are optimal? The answer to this seems to be practice, practice, practice. After a few hundreds of hours, your brain will pickup and learn good moves by just seeing the board pattern.

But to speed up the process and make it more scientific, there should be some general tips about which moves are optimal. This will require some mathematical analysis of the game.

I see 2 ways to practice. One is taking your time to analyze each move, and don't worry about the score. For example, you might make just 5 moves in the whole game. (e.g. spend more than 10 second per move.) The goal of playing slowing is to get you to think and feel the pattern. When you think, you learn better on which moves are better.

You want to maximize making one of those special gems. Because repeatedly getting the multiplier is the only way to score high.

The other pratice you need to do sis speed. For example, if you play for a hour a day, you might do 30 minutes thinking thru each move. Then, for the next 30 minutes gaming, try to make each move as fast as possible.

## Mathematical Analysis on Optimal Moves

much more to be written here … later.

From a math perspective, there's clearly a optimal move. The choice is tough especially when there are lots possible moves.

This is the hard part of this article that requires much research. A true optimal move will probably require a computer analysis of search tree of a given board configuration.

But some rough guide should still be possible. There are many simple questions to be asked. For example, is it better to do vertical or horizontal ones? What's the best way to make a fire gem? What are the general, easy-to-understand, ways to move the gems to places you want?

## Generalization of the Game

This game can be easily generalized. For example, the board can be triangular or hexagonal, and the board topology can torus or other surfaces. For pictures of different grid, see: Go Variations on Tilings. Also, it can be 3-dimensional or higher.

The number of different colors can also change. The fewer the color, the easier the game.

The rule of swapping can also be changed. Bejeweled lets you swap adjacent pieces. But for example, it could allow swapping of diagonal pieces. Or, the swapping could be a particular configuration (e.g. like chess knight's move).

The clearing of pieces can also be made into n-in-a-row, instead of n = 3.

In general, i think this game can be considered as a cellular automata. Such puzzles are basically infinite in variety, and no predictive mathematical theory are possible except of just running thru the game step by step. See: Go Board Game as Cellular Automata.

## References

Here is a couple basic guides on strategy but nothing mathematical. I read them before writing this article.

- Bejeweled Blitz on Facebook: Beginner's Guide to Playing and Getting a High Score (2011-05-24) Written by Finn Orfano, edited By Milawe. @ Source www.brighthub.com
- Advanced Guide for Beating Your Friends in Bejeweled Blitz (2011-05-23) By Written By Milawe, edited by Michael Hartman. @ Source www.brighthub.com