Skip to main content
Links to the University of Cambridge website
Links to the NRICH website Home page
Maths at Home
Summer Challenges 2024
menu
search
Teachers
expand_more
Early years
Primary
Secondary
Post-16
Events
Professional development
Students
expand_more
Primary
Secondary
Post-16
Parents
expand_more
Early Years
Primary
Secondary
Post-16
Problem-solving Schools
About NRICH
expand_more
About us
Impact stories
Support us
Our funders
Contact us
search
Site search
search
Or search by topic
Number and algebra
The Number System and Place Value
Calculations and Numerical Methods
Fractions, Decimals, Percentages, Ratio and Proportion
Properties of Numbers
Patterns, Sequences and Structure
Algebraic expressions, equations and formulae
Coordinates, Functions and Graphs
Geometry and measure
Angles, Polygons, and Geometrical Proof
3D Geometry, Shape and Space
Measuring and calculating with units
Transformations and constructions
Pythagoras and Trigonometry
Vectors and Matrices
Probability and statistics
Handling, Processing and Representing Data
Probability
Working mathematically
Thinking mathematically
Mathematical mindsets
For younger learners
Early Years Foundation Stage
Advanced mathematics
Decision Mathematics and Combinatorics
Advanced Probability and Statistics
Mechanics
Calculus
Tiled Floor
Age
11 to 14
Short
Challenge Level
Secondary curriculum
Problem
Solutions
Answer
: 3025
The sequence of all floors
Even floors: 2 purple tiles per row (total number even)
Odd floors: 1 more purple tile than the previous floor (total number odd)
109 purple tiles
108 = 2$\times$54 so floor length 54 has 108 purple tiles
$\therefore$ floor length 55 has 108 + 1 = 109 purple tiles
Floor length 55 has 55$^2$ = 3025 tiles
The sequence of odd-sided floors
Each odd-sided floor contains the last odd-sided floor in the middle, outlined in blue.
4 purple tiles are added each time
Side length 2 longer each time
109 = 1 + 4$\times$?
= 1 + 108
= 1 + 4$\times$27
So side length = 1 + 2$\times$27 = 55
$\therefore$ total number of tiles = 55$^2$ = 3025
Finding the number of rows
There are 2 purple tiles on each row,
Except for the middle row which has 1 purple tile (on odd-sided floors)
109 tiles = 1 purple tile in the middle
+ 108 purple tiles in pairs
= 1 purple tile in the middle
+ 54 pairs on 54 rows
$\therefore$ there are 1 + 54 = 55 rows
So there are 55$^2$ = 3025 tiles
You can find more short problems, arranged by curriculum topic, in our
short problems collection
.