The River

The River is the 10th level pack and the latest pack added to the game – inserted before Final Exam but after the rest of the packs.  

The River Design 

The pattern for the River designs has 3 sectors.  The middle sector (sector 1) or “the River” is the largest sector.  The size ranges from 3 tiles to 7 tiles high in the center – the above template is 4 tiles high.  

For each design in the River, the two dark sectors are always equivalent in size in shape.  The height of the sectors also range from 3 to 7 tiles high.  

Notice that each of the 3 sectors are connected to the other 2 sectors, and to itself via shadow tiles.

 

Corner Shadow Templates

The corner shadow designs in the River follow one of the above basic templates.  The first template has exactly the same walls as the base template.  The number of corner shadows is 1 less than the height of the dark sectors. 

The second template differs in that 2 walls are removed and the number of corner shadows is equal to the height of the dark sectors.  These are the only designs in the river where the center sector has a vertical shadow tile.  

Board Sizes

The 12 board sizes in The River range from 9×9 to 13×19 (the largest board in the game).   The number of columns is always odd – either 9, 11, or 13 columns.

For each size there is at least 1 design following the standard template, and at least 1 design featuring corner shadows.  Generally the levels alternate between standard templates and the corner shadow version – in total there are 78 levels without corner shadows and 72 with.  

The River Color Scheme

The River features a unique color scheme.  While the 20 colors are exactly the same as the 20 colors in the standard Metro theme used throughout the game, they have been re-ordered to emphasize blue and green, while putting the purple/pink colors to the back.

As a result, the early levels (which have fewer dots due to the smaller board size) feature mostly blue and green dots, while yellow-red colors appear more frequently as you progress through the pack. 

Only a handful of levels reach 15+ pairs of dots – so you rarely see any purple.  

The River Color Groups

Another unique feature is that for each level, the colors of the dots were changed so that the longer lines are of the same tone.  The exception is that regardless of the tone for each level, the dots of this shade of blue are one of the longer lines in the level.  

Here’s how this works:  

Based on the number of dots in the level, one of the above color groups is randomly selected.  The pairs of dots are then sorted in descending order based on the length of the lines connecting them.  The top X dots are then randomly assigned to the selected color group, while the remaining dots are assigned to the other colors.  

One technical detail – shadow tiles count as half when calculating line length – even though they do not count towards the score or fill of a level. 

Here are a couple of examples:  

This is a green level. 

This is an orange level. 

Besides the aesthetic value – this information can be used to help solve each puzzle.  For one – you always know that the line that connects the baby blue dots are generally longer than normal and almost never very short. 

For another – once you see a long line of any shade, you can assume that other lines of that shade are also going to be relatively long.  Conversely, if a short line connects a pair of dots, you can rule out long lines for other dots with the same shade of color.  

The River Bridge

While testing levels for The River, we noticed that a lot of the levels that we liked featured a bridge in the exact center of the board (as shown above).  

So we generated lots of levels that had a bridge in this spot – some with and some without other bridges (in general, our program randomly chooses where to place bridges).  

We thought about and tested an entire pack with bridges in this spot – but ultimately decided that while we liked these levels, it was better to have more variety. 

For example, the placement of the green dots in the above level shows up frequently in these levels.  It’s pretty cool when you encounter it the first few times, but since the line connecting them is almost always the same (it never crosses the bridge, as you might expect), it becomes redundant.  

Still, since so many levels were generated with this bridge, many of them made it into the pack itself.  A total of 39 out of the 150 levels have a bridge in the center, 31 of which it is the only bridge.  82 of the levels have at least one bridge in any spot – of those 43 have only 1 bridge.  

These numbers are by accident and design.  Our program selected the levels for The River without considering bridges at all.  But since the pool of levels included lots of levels with a bridge in this spot, it picked a lot of them.  

Selecting the Designs

A total of 68 different designs were tested and 30 were selected for The River.  

The designs vary in 3 main factors: 

• Size of the board
• Size of the center (conversely, size of the dark sectors) 
• Number of corner shadows

The name of each design is based off these 3 factors.  For example:  the 15×11 5C 4CW Plateau design is 11 columns and 15 rows, has a center that is 5 rows high in the center, and has 4 corner shadows (CW stands for corner warps, which is a synonym for corner shadows).  

We tested board widths ranging from 8 tiles to 13 and heights ranging from 8 tiles to 19.  For most of these sizes we tested both of the corner shadow templates above.  Our testing revealed that odd numbers of columns worked better than even, and that there generally is less of a difference when changing the height.  

This is why you only see 3 different widths in The River, but 10 different board heights.  

Selecting the Levels

Once we decided on the designs to include in the pack, a huge number of levels were generated for each design – about 10,000 per each.  In total, over 300,000 levels were generated for the 165 levels included in the game (150 for The River plus 15 for Final Exam).  

Each level is given a score based on several factors – including average line length, shadows crossed, lines crossing shadows, percentage of filler lines (lines 4 or fewer tiles in length), maximum shadows crossed by a line, etc.  We played and rated hundreds of levels – then used these ratings to generate a weighted linear regression for all of these factors. 

For each design, we included the highest rated level based on this regression (a total of 30 levels out of the 150). while the rest of the levels chosen randomly from a subset of the levels generated.  

Why not just the top rated?  Mostly for variety – the highest rated levels tend to be similar.  The smaller the board size, the more similar they are.  

Thus a significantly higher percentage of the levels generated are considered for the smaller boards.  

For The River, the percentage starts at 13% and goes down to 2%.  This is much lower than most of the other packs in the game (Towers is somewhat close with a range of 15% to 5%).  On average a pack starts at about 50% for the smallest boards and drops to around 10% for the largest boards.  

Statistics

The river has the highest maximum score (49,152) of any pack in the game due to its large board sizes.  The average board size is 164 squares – 11.7 columns and 14.7 rows.  Only Giant Steps (as of the recent update), Towers, Pyramids, and Wide Jags have any levels that are that size or larger. 

Despite the large board sizes the average number of dots per level is only 10.05 – which ranks below Pyramids, Wide Jags, and Towers.  As a result the average line length is exceptionally high at 16.31 – over 2 tiles longer than the next highest pack.  On average 14.35 shadows are crossed – which is 2nd in the game, but the percentage of 54.3%  shadows crossed is dead last (although only 5% below the average). 

The 14.35 shadows crossed breaks down to :

5.73 Vertical Shadows Crossed, 48.8% of total.  The latter is 7% less than the next lowest pack in the game (Giant Steps).  

8.63 horizontal shadows crossed, 58.8% of total.  The former is 2.6 higher than the next highest pack in the game (again, Giant Steps), although the percentage is only 3rd in the game.  

The only other pack in the game where most boards are taller than they are wide is Towers.  Interestingly, Towers has the completely opposite dichotomy of Warps Crossed (even more extreme, even).  It ranks 1st in vertical warps crossed at 81.57% and last in horizontal warps crossed at 36.45%, both by huge margins.  

The distribution of horizontal/vertical shadows crossed for these two packs shouldn’t be surprising – The River has two sets of walls that cross the board horizontally, while Towers has two sets of walls that cross the board vertically.  

It is surprising that the separation is so much smaller for The River than it is for Towers.  We thought this might have to do with the corner shadow levels – so we tested that. 

Vertical Shadows Crossed:  Normal: 5.17/43.9%, Corner: 6.33/54.9%

Horizontal Shadows Crossed:  Normal: 8.9/59.9%, Corner: 8.33/57.5%.  

The difference is in fact significantly higher for levels without corner shadows – but it still doesn’t come anywhere near Towers.  

Conclusion

A lot of thought and effort was put into creating The River.  The pattern itself warranted it – we found the levels to be both interesting and have a lot of variety.  

While each design is based on the same pattern with seemingly slight differences – if there is one thing we’ve learned while creating Connect the Dots:  Shadows, it’s that small differences in designs can make large differences in the levels themselves. 

In The River, the differences between the levels of each of the 30 designs are subtle.  You will see some common patterns in the lines, but they are rarely obvious.  The levels are hard, but not unfair if you are willing to use hint mode at times.  

The River is our new favorite pack in the game.

Pack Stats

Pack Order: 10

Difficulty: 8/10 (rank T2/11)

Max Score:  49152 (rank 1/11))

Board Size:  9×9 to 13×19, 69 to 247 tiles

Board Designs:  30

Avg Dots:  10.05 (rank 4/11)

Avg Shadows/Crossed:  14.35 (rank 2/11) Percentage 54.3% (rank 11/11)

Percent of Lines Crossing Shadows:  57.8% (rank 8/11)

Avg Line Length: 16.31 (rank 1/11)

Line direction changes:  52.8% (rank 7/11)

Avg Max Shadows Crossed By Line:  6.08 (rank 3/11)

Next Pack:  Final Exam