Found problems: 178
For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?
Given a positive integer $n$, an $n$-gun is a $2n$-mino that is formed by putting a $1 \times n$ grid and an $n \times 1$ grid side by side so that one of the corner unit squares of the first grid is next to one of the corner unit squares of the second grid. Find the minimum possible $k$ such that it is possible to color the infinite planar grid with $k$ colors such that any $n$-gun cannot cover two different squares with the same color.
[i]Itf0501[/i]
With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
[asy]
unitsize(0.5 cm);
draw((0,0)--(1,0));
draw((0,1)--(1,1));
draw((2,1)--(3,1));
draw((0,2)--(3,2));
draw((0,3)--(3,3));
draw((0,0)--(0,3));
draw((1,0)--(1,3));
draw((2,1)--(2,3));
draw((3,1)--(3,3));
[/asy]
Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that
- the rectangle is covered without gaps and without overlaps
- no part of a hook covers area outside the rectangle.
A square board with dimensions $100 \times 100$ is divided into $10 000 $unit squares. One of the squares is cut out. Is it possible to cover the rest of the board by isosceles right angled triangles which have hypotenuses of length $2$, and in such a way that their hypotenuses lie on sides of the squares and their other two sides lie on diagonals? The triangles must not overlap each other or extend beyond the edges of the board.
(S Fomin, Leningrad)
You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box.
a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img]
b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?
Crocodile chooses $1$ x $4$ tile from $2018$ x $2018$ square.The bear has tilometer that checks $3$x$3$ square of $2018$ x $2018$ is there any of choosen cells by crocodile.Tilometer says "YES" if there is at least one choosen cell among checked $3$ x $3$ square.For what is the smallest number of such questions the Bear can certainly get an affirmative answer?
A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw?
(I Rubanov)
For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically?
Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.
The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.
For any triple $(a, b, c)$ of positive integers, not all equal, We are given sufficiently many rectangular blocks of size $a \times b \times c$. We use these blocks to fill up a cubic box of edge $10$.
(a) Assume we have used at least $100$ blocks. Show that there are two blocks, one of which is a translate of the other.
(b) Find a number smaller than $100$ (the smaller, the better) for which the above statement still holds.
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.
Let $n\geq 2$ be a natural number. A unit square is removed from a square $n$ x $n$ and the remaining figure is cut into squares 2 x 2 and 3 x 3. Determine all possible values of $n$.
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?
(A Shapovalov)
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).
Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".
Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
Is it possible to tile an $8\times8$ board with dominoes ($2\times1$ tiles) so that no two dominoes form a $2\times2$ square?
A $50 \times 50$ square board is tiled by the tetrominoes of the following three types:
[img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img]
Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling.
(Folklore)
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.)
[img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png
[/img]
Alexandru Girban