THE JOY OF SIX

This puzzle is adapted from my mobile phone game Daily Puzzle, recently published in North America by Sorrent (www.sorrent.com).

1. The correct order is 3-6-2-5-1-4. Book 4 must be next to book 1, and 3 must be next to 6, which means the six books can only be in the order 4-1-5-2-6-3 or (backward)

3-6-2-5-1-4. The answer is 3-6-2-5-1-4, the order with the smallest leftmost number.

2. The correct order is 3-4-2-5-1-6. Book 6 can only be next to book 1. Book 5 can only be next to books 1 or 2. So the sequence 6-1-5-2 (or backward, 2-5-1-6) must be part of the answer, followed or preceded by books 3 and 4 in either order. So the answer with the smallest leftmost number is 3-4-2-5-1-6.

3. The order is 5-3-1-4-6-2. The first book must be 5 because 5 cannot be the second digit of any of the two-digit numbers. The second book must be 3, because all other options yield a two-digit number divisible by 3 or 4. The third book cannot be 4, because the sequence 5-3-4 leaves 6, 2, and 1, which in any combination will create at least one two-digit number that is evenly divisible by 3 or 4. So 5-3-1-4-6-2 is the only sequence that satisfies the condition.

WHAT LIES BENEATH

1.  The numbers that aren’t necessary to solve the puzzle appear in blue.

2.  Four is the greatest number of squares you can touch without uncovering the treasure.

3.  Two squares are always enough. First touch a corner square. The number will reveal on which diagonal the treasure lies. (A 2 in the top left corner square would show that the treasure is on the diagonal that runs from the third square in the top row to the third square in the first column.) Then touch a square at one end of the diagonal. This will tell you where along the diagonal the treasure is hidden.

LINES OF REASONING

This puzzle is adapted from my 365 Brainteasers Page-A-Day calendar for 2005 (Workman Publishing, www.workman.com).

1.  If each red dot is part of the loop, then each red dot must have two black lines coming out of it. So the two y lines in the lower left corner must both be black because they are the only two lines that touch the corner dot. (This applies, of course, to all four corners of the grid.) The other two y lines must be black because the dot that is common to both cannot be connected any other way without creating an “illegal” three-way junction—illegal because a three-way junction would not allow for a continuous loop through the puzzle.

2.  The upper n line cannot be black, because it would then create a three-way junction. The lower n line cannot be black because, as mentioned above, there must be two black lines coming out of the bottom right corner dot—filling in the lower n  line would form a separate square loop.

3. 

4.  Glenn Iba recently found a grid with just eight lines that forces a unique solution.

Is there a solution with fewer lines? No one knows (yet).