Logic
# Logic Warmups

$\frac{1}{4}$ the radius of the larger circle. The smaller circle rolls around the larger circle for one trip, back to its starting point, without any slipping.

The radius of the smaller circle isHow many times would the red dot on the smaller circle rotate around the center of the smaller circle?

$\begin{array} { l l l l l l } & S & E & V & E & N \\ & S & E & V & E & N \\ +& & & S & I & X \\ \hline \\ T & W & E& N&T&Y\\ \end{array}$

Each letter represents a distinct digit.

What is the value of $W$?

You have to cross a large desert covering a total distance of 1,000 km between Point A and Point B. You have a camel and 3,000 bananas. The camel can carry a maximum of 1,000 bananas at any time.

The camel eats the bananas at a constant rate of one banana per kilometer travelled. What is the maximum number of uneaten bananas (rounded off to the closest whole number) that the camel can transport to the other end of the desert?

On the face of it, it seems that you can’t transport even a single banana uneaten to the other end of the desert. However, if that would have been the case, I wouldn’t have posted it and wasted your time.

Hint: You have to make multiple trips back and forth as you transfer all 3000 bananas. Find the most efficient way.

You have a (solid) blue cube and an unlimited number of (solid) red cubes, all of which are of the same size. What is the largest number of red cubes that can touch the blue cube along its sides or parts of its sides?

Touching along edges or at corner points does not count.

You are trapped in a small room with 4 walls. Each wall has a button that is either in an OFF/ON setting though you have no way of telling what the setting is.

When you press a button, you change its setting. If you can get all the buttons to have the same setting i.e. either all four are OFF or all four are ON, you are immediately set free.

In each move, you can press either two buttons simultaneously or just one button. As soon as this occurs, if you haven't been set free, the whole room spins around you violently, leaving you completely disoriented so that you can never tell which side is which.

The starting position is completely at random (except not all four OFF or all four ON). Given any and every possible scenario, using optimal strategy, what is the least number of Moves needed to unquestionably guarantee escape from the room?