Logic? No Problem!

What’s your problem-solving style—brute force or careful reasoning? Either approach may work, but as this month’s collection of puzzles demonstrates, you’ll usually get the answer more quickly and elegantly if you use logic to break things down.

 

THE JOY OF SIX

Six books sit side by side on a shelf. Put the books in order so that the sequence of the numbers on their spines satisfies each of the following conditions.


1. [Easy] Rearrange the books so the numbers of adjacent books differ by at least three and the number of the leftmost book is as small as possible.

2. [Easy] Rearrange the books again, this time so adjacent book numbers add up to no more than seven and the number of the leftmost is as small as possible.

3. [Challenging] Rearrange the books yet again so that adjacent book numbers form two-digit numbers that cannot be divided evenly by 3, 4, or 5. For instance, the order 1-2-3-4-5-6 fails on several counts: The two-digit number 12 divides evenly by 3 and 4; the number 45 divides evenly by 3 and 5; and 56 divides evenly by 4.

WHAT LIES BENEATH

Math educator and researcher Tom O’Brien developed a computer game called Treasure Hunt to teach logical thinking skills to elementary school students. The goal is to find a precious gem that is hidden underneath one of 16 squares in a four-by-four grid. When you touch a square of the grid, a number appears revealing the shortest distance from the square to the treasure as measured in horizontal and vertical steps. For example, the numbers in the grid above show the distance to the treasure from four different squares. (Each arrow represents one step.) There can be several different paths of equal length from a particular square to the jewel.

1.  [Easy] Is there enough information provided to find the treasure in each of the following grids? If so, where is the treasure? If not, where might the treasure be? And if there is more than enough information provided, which numbers are unnecessary?

2.  [Challenging] What is the greatest number of squares you can touch and still not be able to determine for certain where the treasure is?

3.  [Challenging] What is the best strategy for finding the treasure? In other words, how few squares can you touch and still be sure to find the treasure?

LINES OF REASONING 

 

In the grid at right, constructed by Glenn Iba, find a single closed path that passes through every red dot and includes all of the black lines. You can deduce the solution through logic. To set off on the right path, first try answering these questions.

1. [Easy] Grid lines marked y must be black. Why?

2. [Challenging] Grid lines marked n must not be black. Why?

3. [Difficult] Complete the path, which must be a single closed loop that includes every red dot and all of the black lines. The solution is unique.

4. [Unsolved] The 12 black lines drawn into the grid at left force a unique solution. If you were starting this puzzle with an empty grid (without any black lines drawn in), what would be the minimum number of black lines required to force a unique solution?

Bogglers Solutions