Puzzle Tweeter

Consider the 7x7 grid shown below:

Each cell must be filled in with an integer from 1 to 7 that represents the height of the building. The numbers given at the sides (top, bottom, left, right) represent how many buildings are visible when seen from that location. A taller building blocks the view of all shorter buildings behind it. Also, you want each row and column to contain all the numbers from 1 to 7 exactly once. Can you solve the puzzle?

-via The Eighth World Puzzle Championship. A discussion can be found in Tanya Khovanova's blog

The Peg Solitaire puzzle is also known as Conway's Soldiers Puzzle. It consists of a checkerboard that extends infinitely, a horizontal dividing line, and an arbitrary number of pegs (soldiers) below the horizontal line. See the figure below:

A move consists of one peg jumping over another adjacent peg into an empty cell either vertically or horizontally, but not diagonally. The peg that is jumped over is removed after the move.

The goal of the game is to get one peg as far above the horizontal line as possible.

Show that no matter what you do, it is impossible to reach more than 4 levels above the horizontal line when you start with a finite number of pegs.

Hint: Define a potential energy function W(x,y) whenever a peg is present in position (x,y). Define it in such a way that a valid move will always cause the total energy to decrease. Then use conservation of energy to show that a peg at 5 levels above the horizontal line has too much energy.

This puzzle is taken from Wikipedia and is referenced in Pratik Poddar's blog.

Let f(t,n) denote the maximum number of connected regions into which an n-dimensional space can be divided using t hyperplanes that are (n-1)-dimensional. Assume t>=n. We solved for f(t,2) here and f(t,3) here .

Find a recursive relation for f(t,n).

Are you surprised that the answer f(t,n) is the same as the one you obtained in the egg dropping experiment posted here yesterday with t tries and n eggs?

--- That the solutions are same was pointed out by Paul Cuff.

Suppose you have n eggs and a maximum of t tries for the egg drop experiment from yesterday (found here). Let f(t,n) be the maximum number of floors for which it is possible to compute the egg-breaking floor.

Find a recursive relation to compute f(t,n).

- via Wikipedia.

Suppose you have 3 identical sturdy eggs. You are living in a 100 floor building. You like to know the smallest floor from which an egg will break when dropped. However it is time consuming to keep going to each floor for a trial. You want to minimize the worst case number of trials. So you don't want to go from the bottom floor to the top floor one at a time. What is the minimum number of trials needed in the worst case to determine the exact height at which an egg breaks?

What is the maximum number of regions into which 3-d space can be split using n planes?

For example, n=1 plane creates 2 regions, n=2 planes creates 4 regions in space, etc. Refer to yesterday's puzzle here for an easier version.

What is the largest number of regions into which n lines can divide a plane?

For example, 1 line creates 2 regions, 2 lines creates 4 regions, 3 lines creates 7 regions, etc.

-- via discussions in Thomas Cover's puzzle meetings.

A boat carrying a stone is floating in a swimming pool. What would happen to the level of water in the pool if you take the stone from the boat and throw it into the pool?

via Math exchange

Suppose you have a rectangle of dimensions N x 2. (length: N, Width: 2). How many circles of diameter 1 can be embedded in this rectangle so that the circles are non-intersecting?

via stack exchange

5 cars were travelling on a freeway. Each car stopped at exactly 2 red lights. Also each pair of cars stopped together at the same red light at some point. Show that at some instant at least 3 of the 5 cars stopped simultaneously at the same red light.

via AMS math society puzzle corner