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witzar

Member Since --
Offline Last Active Yesterday, 12:58 PM
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Topics I've Started

Covering perforated hexagon with triminoes

14 November 2014 - 05:46 PM

This puzzle is inspired by Covering triangles with triminoes posted by bonanova.

Again we work on a hexagonal tiling of a plane, and the question is about
possibility of covering some shape with triminoes.
Trimino is a "triangle" formed by three unit hexagons sharing common vertex.

The shape to cover is defined as follows:
Let's pick a unit hexagon and call it H1.
Now we recursively define Hn+1 as a sum of Hn and all unit hexagons adjacent to Hn.
So basically Hn is a "hexagon" with side of length n (unit hexagons).

Let Dn be Hn with one unit hexagon at it's center removed.
So, can you cover D2015 with triminoes?


Removing pawns - the game

25 October 2014 - 12:41 PM

Here is the simple game I've invented (if someone invented it before, then I'm not aware of it):

 

A pawn is placed on every square of m*n chessboard.
Two players take alternate turns removing pawns.
On each turn, a player removes one or more pawns.
All pawns removed in a single turn have to be taken from the same row or the same column.
The player who cannot make a move loses (alternatively: the player who takes the last pawn wins).

Let's call the player who begins the "first player" and the other one the "second player".

Which player (the first or the second) has a winning strategy depending on the chessboard dimensions?
 

PS The game can be played on more interesting boards. Just draw some number of crossing lines

on a sheet of paper and place a pawn on each point of intersection of the lines.

The pawns removed in a single turn should come from a single line.

I've tried to play this game on the pentagram (the star: five lines, ten points/pawns) with my 9 years old daughter.

It was fun. Determining which player wins was even bigger fun.