Problems

While walking through the woods in Wonderland, Alice met three very peculiar hunters. They hunted a hare, which was hiding in one of the vertices of the cube \(ABCDEFGH\).

The three hunters fire simultaneously to hit the vertices of the cube (the hunters are all excellent shooters). If they don’t hit the hare, the hare runs over one of the three adjacent edges to the next vertex and hides there. The hunters ask Alice to help them. They want to shoot the hare firing not more than 4 times, but not sure how to do it. Can you help Alice advise the hunters? (please write four vertex triples to be fired by the hunters).

In the middle of the Dark Forest in Wonderland there is a large square clearing, where a wolf is sitting right is the middle of the square, and four dogs are sitting at the four vertices of the square. The wolf can run inside the square with maximum speed \(v\), while the dogs can run along the edges of the square with the speed \(1.5v\). It is known that the wolf kills a dog if they meet one to one, and two dogs kill the wolf if they overpower it together. Can the wolf escape from that square into the forest?

You have two sticks and matchbox. Each stick takes exactly an hour to burn from one end to the other. The sticks are not identical and do not burn at a constant rate. As a result, two equal lengths of the stick would not necessarily burn in the same amount of time. How would you measure exactly 45 minutes by burning these sticks?

One of your employees insists on being paid daily in gold. You have a gold bar whose value is that of seven days’ salary for this employee. The bar is already segmented into seven equal pieces. If you are allowed to make just two cuts in the bar, and must settle with the employee at the end of each day, how do you do it?

You have a 3-quart bucket, a 5-quart bucket, and an infinite supply of water. How can you measure out exactly 4 quarts?

Suppose you had eight billiard balls, the recruiter began. One of them is slightly heavier, but the only way to tell is by put-ting it on a scale against the others. What’s the fewest number of times you’d have to use the scale to find the heavier ball?

How many times a day do a clock’s hands overlap?

How do you cut a rectangular cake into two equal pieces when someone has already removed a rectangular piece from it? The removed piece can be of any size or orientation. You are allowed just one straight cut.

Why does a mirror reverse right and left instead of up and down?

Remove a \(1 \times 1\) square from the corner of a \(4 \times 4\) square. Can this shape be dissected into \(3\) congruent parts?