Pentomino Solutions 6×10 Rectangle
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Solving the Puzzle: The 6×10 Pentomino Challenge (with a Big Help)
The 6×10 pentomino challenge asks you to fit all 12 pieces into a rectangle perfectly—no overlaps, no gaps, just pure geometric magic. There are 2339 unique solutions! My go-to solution site has been isomerdesign.com. In this post, I’m not solving the 6×10 challenge myself, but I’ll visualize some solutions using R & trusty ggplot2.
Pakcages Used in This Blog Post
library(tidyverse) # Easily Install and Load the 'Tidyverse'
library(cowplot) # Streamlined Plot Theme and Plot Annotations for 'ggplot2'
library(sf) # Simple Features for R
library(patchwork) # The Composer of Plots
### Just playing around with color palette
retro_col_a <- c("#00A0B0", "#6A4A3C", "#CC333F", "#EB6841", "#EDC951")
retro_col_b <-c("#CA0B0B", "#EAA109", "#71A6AE", "#18668C", "#06394D")
retro_col_c <- c("#325A64", "#44838F", "#68D0BD", "#F53F19", "#891C29")
retro_col_d <- c("#241965", "#653993", "#9F4094", "#B73D6E", "#F19406")
retro_col_e <- c("#383431", "#79C39E", "#EAD1B5", "#EE9B69", "#E77843")
retro_col_f <- c("#8E2605", "#E54B1F", "#FDC018", "#628A81", "#5F3924")
retro_col_g <- c("#811638", "#0B7978", "#FCB632", "#F27238", "#C32327")
retro_col <- c(retro_col_b, retro_col_e, retro_col_a, retro_col_d, retro_col_c, retro_col_f, retro_col_g)
I started by loading few useful pakages above. tidyverse makes data wrangling simple, while sf helps manage spartial geometries. Additionally I’ve prepped retro-inspired color palettes so that I can give the visuals a nostalgic puzzle game vibes.
Loading the Solutions
Reading Text Solution File
### Thank you to solution as text!
sol_df <- read_csv("https://isomerdesign.com/Pentomino/6x10/solutions.txt",
col_names=F)
### 2339 Solutions included in CSV
### Turn solution into data frame
make_coord_df <- function(x){
#x <- sol_df$X2[[1]]
x <- str_split(str_remove_all(x," "),"",simplify=T)
matrix(x,nrow=6,ncol=10, byrow=T) |>
as_tibble() |>
mutate(y=row_number()) |>
pivot_longer(-y) |>
mutate(x=as.integer(str_extract(name,"\\d+"))) |>
select(x,y,piece=value) |>
group_by(piece) |>
mutate(x_min=min(x),y_min=min(y),
x_max=max(x),y_max=max(y),
idx=row_number(x+y),
adj_x=x-x_min+1,adj_y=y-y_min+1) |>
ungroup()
}
### df for data frame
sol_df <- sol_df |>
mutate(solution_df = map(X2,make_coord_df))
### df_long for data frame that's unnested
sol_df_long <- sol_df |>
unnest(solution_df)
The solutions are provided in a plain text file, where each solution is represented by 10 strings of characters x 6 rows for each of 2339 solutions for 6x10 rectangle. Each letter corresponds to one of the 12 pentomino pieces. Using this as input, I created a function to process each solution into a structured tibble (data frame).
Convert to SF Object
# Function to turn coordinate as center and conver to square
create_square_fence <- function(x, y) {
st_polygon(list(matrix(c(
x-0.5, y-0.5, # Bottom Left
x+0.5, y-0.5, #Bottom Right
x+0.5, y+0.5, #Top Right
x-0.5, y+0.5, #Top Left
x-0.5, y-0.5 # Close the polygon by coming back to bottom left
), ncol = 2, byrow = TRUE)))
}
### Convert data frame into sf object
sol_sf <- sol_df_long |>
rowwise() |>
# For each row, create a square geometry from the x, y coordinate
mutate(geometry=list(create_square_fence(x,y)),
geometry_adj=list(create_square_fence(adj_x,adj_y))) |>
ungroup() |> # Remove rowwise grouping
group_by(X1,piece) |>
summarise(geometry=st_union(st_sfc(geometry)),.groups="drop",
geometry_adj=st_union(st_sfc(geometry_adj),.groups="drop")) |>
st_sf()
pieces_63 <-sol_sf |>
st_drop_geometry() |>
count(piece,geometry_adj,sort=T) |>
arrange(piece) |>
mutate(i=row_number()) |>
mutate(col=colourvalues::color_values(i,farver::decode_colour(retro_col)))
sol_sf_comb <- sol_sf |>
inner_join(pieces_63 |> select(col,geometry_adj,piece))
Once the data frame was ready, next step was to convert it into sf geometries for spatial analysis and visualization. Here’s how I’ve tackled it.
- Square Polygons: I wrote a small helper function to create a square polygon for each (x,y) coordinate. Each square represents a single unit of the pentomino square pieces.
- Merging into Shapes: To represent entire pentomino pieces, I’ve combined all individual square polygons for a piece into a single shape using st_union().
- Normalized Geometry for Variations:
- Pentomino pieces can appear in up to 8 variations (rotations and flips). For example, a single piece might have multiple “faces”, depending on how it’s oriented in the solution.
- To identify which variations of a piece was used in each solution, I created adjusted geometries by aligning the shapes relative to their bounding boxes.
Heatmap of Pentomino Placement Patterns
With the 2339 Solutions processed and converted into geometries, I decided to analyze where each pentomino piece tends to appear on the board. The result is the heatmap you see below.
The code using geom_tile() to represent the frequency of each piece at each (x,y) position across all solutions. A facet is created for each pentomino piece, so we can see the pattern differences.
Heatmap of Pentomino Placement Patterns
sol_df_long |> count(x,y,piece) |> ggplot(aes(x=x,y=y)) + geom_tile(aes(fill=n)) + coord_fixed() + facet_wrap(~piece) + scale_fill_viridis_c(option="F",trans="sqrt","apperence") + theme_map(font_family="Roboto Condensed")
Insights from the Heatmap
- X Stays Central: The X-piece avoids edges entirely, sticking to the middle where it fits best.
- I Loves the Left Edges: The I-piece often hugs the edges, but there are a few spaces it avoids altogether.
- F and W Are Versatile: These flexible shapes show up all over, but slight tendency towards the center
- V and Y Favor the Edges: Their hook-like shapes are perfect for corners and boundaries.
- T Favors the Top Left: The T-piece seems to favour the top left-corner according to the heatmap.
- Z Leans Asymmetrically: Z leans to one side, and more likely to stay away from the edges.
- L Sticks to Edges: These elongated shapes often sit along the board’s boundaries, especially the bottom left.
- P and U Fits Everywhere: the P-piece slots into difference places.
Recapping the Shapes
Piece Shapes Recap
p_sep <-pieces_63 |>
group_by(piece) |>
mutate(j=row_number(i)) |>
st_sf() |>
ggplot() +
geom_sf(aes(fill=I(col)),color="#fffff3") +
facet_grid(j~piece) +
theme_nothing()
#theme_minimal()
p_stack <-pieces_63 |>
group_by(piece) |>
mutate(j=row_number(i)) |>
st_sf() |>
ggplot() +
geom_sf(aes(fill=I(col)),color="#fffff3") +
geom_sf_text(aes(label=piece,
geometry=geometry_adj+c(0,3)),
family="American Typewriter", size=5,
data = . %>% filter(j==1)) +
facet_grid(~piece) +
theme_nothing()
p_sep + p_stack + plot_layout(ncol=1, heights=c(4,1))
In above I created a visualization to recap the 12 pentomino shapes, each column shows different variations of piece faces. There are 12 free pieces, but 63 fixed pieces.
Some Solutions !
Since it’s bit hard to display all the solution at once… I’ve first chosen these 12 special solutions. These 12 solutions are particularly interesting because they belong to 3 unique sets where the pieces remain in their default orientation—no flipping or rotation is required! Within each set, there are 4 solutions that differ only in the arrangement of the pieces on the board.
Special 12 Solutions (3 Sets x 4 Solutions)
sol_by_color <-sol_sf_comb |> st_drop_geometry() |> group_by(X1) |> summarise(col_pal=paste(sort(col),collapse=",")) |> ungroup() |> add_count(col_pal) |> arrange(desc(n),col_pal) sol_sf_comb |> inner_join(sol_by_color) |> filter(n==4) |> ggplot() + geom_sf(aes(fill=I(col))) + facet_wrap(~X1) + theme_map(font_family="Roboto Condensed")
Another Special 9 Solutions (3 Sets x 3 Solutions)
sol_sf_comb |> inner_join(sol_by_color) |> filter(n==3) |> arrange(col_pal) |> ggplot() + geom_sf(aes(fill=I(col))) + facet_wrap(~fct_inorder(factor(X1))) + theme_map(font_family="Roboto Condensed")
180 Solutions where Pair Exists
sol_sf_comb |> inner_join(sol_by_color) |> filter(n==2) |> arrange(col_pal) |> ggplot() + geom_sf(aes(fill=I(col))) + facet_wrap(~fct_inorder(factor(X1)),ncol=18) + theme_nothing(font_family="Roboto Condensed")
Wrapping Up 🎁
Visualizing these solutions has been like uncovering hidden patterns in a puzzle. It’s been pretty fun execise learning to utilize some of function in sf packages too. And most importantly, now I have solution on my blog I can put away the actual puzzle board after the play.
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