Marble Selection [No. 692]
By placing 1 red marble in a jar and the remaining 49 red marbles with all 50 blue marbles you maximize the chance of randomly selecting a red marble. There is a 50% chance of selecting from either jar. The one with a red marble has a 100% chance of red. The other jar has a 49/50 ~ 50% chance of red. Therefore the total probability is 50%(100%) + 50%(≈50%) ≈ 75%.
Lions and Bears and Logic, Oh My! [No. 431]
When you solve this the case of only one lion and one sheep, it is clear that the lion would eat the sheep as there is no threat to be eaten. When you solve for the case of two lions and one sheep, it is clear that neither lion would eat the sheep, because if they did, we would have the case of one lion and one sheep which we know results in being eaten. This problem becomes a repeating even/odd lion pattern. Here, with 9 lions, it is safe for a lion to eat the sheep, because it will be safe once there are 8 lions and 1 sheep.
Brain Power [No. 943]
Since the camper in the back has no chance of knowing their hat color, she can signal the rest of the group. Let’s say the plan is that she says “Red” to mean there are an even number of red hats in front of her, and “Blue” means an odd number. Then the next camper in line can take that information together with if they see an even or odd number of red hats to figure out what color hat they are wearing. Each successive camper can do the same and be spared. The optimal solution saves everyone and gives the camper at the back of the line a 50% chance. Not bad!
Bootlegger's Advice [No. 103]
The server’s plan was to drink weak moonshine before the meeting and prepare water as his moonshine. In that case, the bartender’s moonshine would serve as the antidote and he would just be chasing it with water, his drink. While the bartender would drink water and then her own moonshine and surely go blind.
The bartender, realizing this plan, decided to also provide water. Now she just drinks water twice, while the server ends up blind from the moonshine he takes before the meeting, having only water to drink twice afterwards!
The king has nothing but water and never gets the strong brew he seeks.
Cheater, cheater, test deceiver [No. 423]
There are two scenarios to consider:
The first two hats in the lineup are the same color. In this case the last student knows his hat is the opposite color.
The first two hats in the lineup are different colors. The student in the middle, not hearing the last person in the line respond (as he cannot be certain of his hat color), knows that his must be a different color than the hat he sees of the first student in line.