CTK Insights

This is just to document a few simple math and logic activities recently added to the Interactive Mathematics Miscellany and Puzzles site.

Filling a Grid with Good Neighbors

There is a number of chips placed in the squares of an N×N grid. We can add a chip to a square, provided it has at least two occupied 4-neighbors. The task is to position the chips initially so as to be able to fill the whole grid in a number of steps.

The originally occupied squares are colored in orange and display 0 in the middle. Other squares show the number of the step at which they were added.

What is the least number of squares to be occupied initially to ensure that at the end there are no empty squares? How to position the chips initially to make the process last as long as possible?

Flipping Pancakes

"Pancakes" (all of different sizes) are stacked in a random order on top of each other.

A flip consists in choosing an integer and turning upside down that number of top pancakes. You are allowed to flip simultaneously a number of top pancakes. The number of the pancakes to flip is determined by the size of the very top pancake. You are to observe and subsequently to prove that the process eventually terminates and explain why.

A Game of Candy Squares

This two-player game starts with two heaps of candy squares. To make a move, a player gobbles one heap and divides the other one into two. The players take turns; the one who can't move, i.e., divide a heap, loses the game. Is there a strategy to win the game?

Treasure Hunt in a Square Grid

The so-called taxicab or city-block distance is at the heart of a computer game "Treasure Hunt" developed by the late math educator and researcher Tom O'Brien, also known under the nom de plume of Professor Tobbs. The game is to teach elementary school students logical thinking. The goal of the game is to find a treasure hidden under one of the squares in a N×N grid. The way to locate the treasure is by touching a square on the grid. If you are lucky to stumble on the square that conceals the treasure the game stops. Otherwise, you are given a clue - the taxicab distance from the selected square to the treasure. You may touch as many squares as you need to solve the puzzle.

Treasure Hunt in a Square Grid From Outside the Square

This a modification of the previous puzzle. The idea is to teach elementary school students logical thinking. The goal of the game is to find a treasure hidden under one of the squares in a N×N grid. The way to locate the treasure is by touching a square on the grid. If you are lucky to stumble on the square that conceals the treasure the game stops. Otherwise, you are given a clue - the taxicab distance from the selected square to the treasure. You may touch as many squares as you need to solve the puzzle.

In the modified version you touch the regions outside the grid, one step away so to speak. I believe this modification is more suggestive of why touching a corner square reveals a diagonal that contains the treasure, although the reason is pretty simple anyway.

Skyscrapers Puzzle

Imagine a well designed city where all streets are either parallel or perpendicular. The city buildings are arranged in a N×N square of peculiar property: the numbers of floors in the buildings form a Latin square. These are not shown. It is your task to retrieve those numbers.

You are given clues: the numbers on the sides of the N×N square show how many buildings are seen from that point in the corresponding (either horizontal or vertical) direction. To produce a number click repeatedly in a square where a building is expected.

Subtraction in a Rectangular Array

N×M non-negative integers are arranged in N rows and M columns. At any step you may subtract the same number from any two adjacent cells. Is it always possible to reach the state where all the displayed integers are zero?

There are some necessary conditions for the puzzle to be solvable, but the selection of moves is crucial. Below, one gets in two steps from the left position (which could be solved) to the right one which is unsolvable.

Related posts:

1. Euclid's Game