We have a $3 \times 3$ square grid, and we must color $5$ squares in this grid, but each colored square must be connected to all other colored squares (There must be one connected shape, not multiple shapes). How many different ways can we color this grid? Thanks in advance.
2026-04-24 22:30:38.1777069838
How many different shapes that consist of five bordering squares can there be in a $3 \times 3$ grid?
237 Views Asked by user307244 https://math.techqa.club/user/user307244/detail At
1
There are 1 best solutions below
Related Questions in COMBINATORICS
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- The function $f(x)=$ ${b^mx^m}\over(1-bx)^{m+1}$ is a generating function of the sequence $\{a_n\}$. Find the coefficient of $x^n$
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Hard combinatorial identity: $\sum_{l=0}^p(-1)^l\binom{2l}{l}\binom{k}{p-l}\binom{2k+2l-2p}{k+l-p}^{-1}=4^p\binom{k-1}{p}\binom{2k}{k}^{-1}$
- Algebraic step including finite sum and binomial coefficient
- nth letter of lexicographically ordered substrings
- Count of possible money splits
- Covering vector space over finite field by subspaces
- A certain partition of 28
- Counting argument proof or inductive proof of $F_1 {n \choose1}+...+F_n {n \choose n} = F_{2n}$ where $F_i$ are Fibonacci
Related Questions in PUZZLE
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- Number of divisors 888,888.
- Who has built the house of Mason?
- Is there any tri-angle ?
- In what position , the dogs will reside?
- Number of ways to go from A to I
- Who is the truth teller (logic puzzle)
- How many solutions are there if you draw 14 Crosses in a 6x6 Grid?
- Symmetric latin square diagonal elements
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
OK, so the shapes are all 'pentominoes':
https://en.wikipedia.org/wiki/Pentomino
The pentominoes that fit into a 3x3 square are:
F,P,U,T,V,W,X,Z
Assuming rotation and mirroring can all amount to different colorings, we thus have the following possibilities:
F: 8 possibilities (mirror is different, and each of the four rotations is different)
P: 16 possibilities (mirror is different, and each of the four rotations is different, and it can be put in two different pairs of columns/rows)
U: 8 possibilities (each of the four rotations is different, and it can be put in two different pairs of columns/rows)
T: 4 possibilities (just 4 rotations)
V: 4 possibilities (just 4 rotations)
W: 4 possibilities (just 4 rotations)
X: 1 possibility
Z: 4 possibilities (2 rotations + mirror)
For a total of: 49 possibilities