NPR Sunday Puzzle (Aug 17, 2014): Target Practice

NPR Sunday Puzzle (Aug 17, 2014): Target Practice:

Q: You have a target with six rings, bearing the numbers 16, 17, 23, 24, 39, and 40. How can you score exactly 100 points, by shooting at the target.

Finally a math puzzle, but then it has to be one that isn't very challenging. If you are having trouble, just keep firing arrows; you'll hit it eventually.

Edit: You'll need the full complement of 6 arrows.

A: 16 + 16 + 17 + 17 + 17 + 17 = 100

NPR Sunday Puzzle (Mar 3, 2013): Dinner Party Musical Chairs

NPR Sunday Puzzle (Mar 3, 2013): Dinner Party Musical Chairs:

Q: Eight people are seated at a circular table. Each person gets up and sits down again — either in the same chair or in the chair immediately to the left or right of the one they were in. How many different ways can the eight people be reseated?

For this puzzle, I think we have to assume each seat position and person is unique. Also, I assume Will wants seating arrangements where each person has their own chair (no sharing). What I don't see, is why the table has to be circular. Couldn't it be square and we could still figure out how to move left or right?

Edit: The first case that might get overlooked is everyone returning to their original seat. The next two cases are where all 8 people move clockwise or counter-clockwise one seat. There can't be any other cycles involving more than two people because that would require someone to move more than one seat, so the remaining cases involve neighboring "couples" swapping seats while others stay still. All that is required is to enumerate the ways to swap couples.

A: There are 49 ways that 8 people could stand up and be reseated (link to PDF containing diagrams). Incidentally, the Online Encyclopedia of Integer Sequences has the answers for various table sizes (A0048162 = 1, 2, 6, 9, 13, 20, 31, 49...) which confirms the answer for 8 people is 49 ways.

NPR Sunday Puzzle (Oct 7, 2012): Hexagon Diagonals - Count the Triangles

NPR Sunday Puzzle (Oct 7, 2012): Hexagon Diagonals - Count the Triangles:

Q: Draw a regular hexagon, and connect every pair of vertices except one. The pair you don't connect are not on opposite sides of the hexagon, but along a shorter diagonal. How many triangles of any size are in this figure?

The diagram in the upper right should help. I've removed one diagonal. It looks like a cool cube, don't you think?

Edit: The words "cool cube" were a double hint. First, the diagram I drew reminded me of the isometric cubes in the Q*Bert video game which was released in 1982. Additionally, if you cube the answer (82^3) you get 551368. I think the result is cool because 55+13=68. As a final clue, in several of my comments, I used the word "lead" which happens to be 82 on the Periodic Table of Elements.

A: 82 Triangles - be sure to watch the video for an explanation of the answer.

GeekDad Puzzle of the Week: Waffle Cuts

GeekDad Puzzle of the Week: Waffle Cuts:

Q: If we only cut along the ridges of a circular waffle, and if each cut traverses the waffle in a straight line from edge to edge, how many different ways can the waffle be cut?
Note: rotations, horizontal flips, and vertical flips of a set of cuts should only be counted once.

Given that there are 6 places to cut vertically and 6 places to cut horizontally, that's a total of 12 cut lines. If you allow for any combination of these 12 lines to be cut or not, you have a total of 2^12 = 4096 ways to divide the waffle. But of course, the puzzle asks for the number of unique ways to cut the waffle, not including any mirrored or rotationally symmetric sets of cuts.

After the official answer to the puzzle is posted, I'll post my solution here.

Edit: The solution is posted, but just the number without any detail. Also, I disagree with their counting of the "no cuts at all" solution as one of the ways to "cut" the waffle. In any case, a full detailing of my solution along with an enumeration of all 665 (or 666) ways to uniquely cut the waffle can be found in Blaine's Solution to the GeekDad Waffle Puzzle.