1. The Coin Conundrum:
You have 12 coins – some are heads, and some are tails. However, all the heads coins weigh the same, and all the tails coins weigh the same, but a heads coin weighs differently from a tails coin. Using only a balance scale three times, can you determine how many heads and how many tails coins you have? (Think strategically!)
2. The Towering Challenge:
Imagine you have 64 colored discs, half red and half blue. You can only stack the discs on top of each other if they are the same color. What is the minimum number of moves required to create two separate stacks, one with all the red discs and the other with all the blue discs? (Think logically!)
3. The Divisibility Dilemma:
Pick any three-digit number (don't tell us!). Now, add the digits of your chosen number together. If the sum is divisible by 3, the original number is also divisible by 3. Is this statement always true? Why or why not? (Explore patterns and divisibility rules!)
4. The Tricky Train:
A train travels 72 miles in the first hour and 60 miles in the second hour. However, due to strong winds slowing it down, it only travels 55 miles in the third hour. If the train continues at this reduced speed, how many miles will it travel in 5 hours? (Beware of assumptions – read carefully!)
5. The Pizza Puzzle:
You and a friend order a large pizza with 8 slices. You agree to split it fairly, but your friend doesn't like olives, which are on half the pizza. How can you cut the pizza into 8 slices, ensuring you both get an equal share with and without olives? (Think creatively and visually!)
1. The Coin Conundrum:
Answer:
- Weigh two coins against each other.
- If they weigh the same, you know all the coins have the same weight (either all heads or all tails). In this case, any two more weighings will reveal how many of each type you have (e.g., weigh 3 coins vs. 3 coins, then 2 vs. 2).
- If they are different, the heavier one is heads (since heads weigh differently from tails), and the lighter one is tails.
- Now, weigh three heads coins together.
- If they weigh the same as two of the original coins weighed together, then there's one tails coin. The remaining coins are all heads.
- If they weigh more than two of the original coins, then there are two tails coins. The remaining coins are all heads.
2. The Towering Challenge:
Answer: Minimum 7 moves.
- You need to move each disc at least once to separate the colors. Since you can only move one disc at a time, the minimum number of moves required is the total number of discs (64).
- However, some moves can be combined strategically.
- Move a red disc to an empty space.
- Now, you can move three blue discs on top of it in one go (since the bottom disc doesn't matter for stacking blue discs).
- Repeat this process (move a red disc, then stack 3 blue on top) until you have all the blue discs stacked.
- The remaining red discs will automatically be in their own stack.
- This method uses 64/4 = 16 moves to stack the blue discs, plus one move for each red disc (which needs to be moved twice – once to create a space and again to stack on top of the blue pile). Therefore, the minimum number of moves is 16 + 2 = 18.
3. The Divisibility Dilemma:
Answer: False.
- This statement only applies if the original three-digit number is a multiple of 3.
- Let's explore a counter-example: Choose 102. The sum of the digits is 3 (1 + 0 + 2). However, 102 is not divisible by 3.
4. The Tricky Train:
Answer: 167 miles.
- The key is to recognize that the question asks about the speed for the 5th hour, not a continuous travel time for 5 hours.
- Since we only know the speed for the first 3 hours (72 mph, 60 mph, and 55 mph), we can't calculate the total distance traveled in 5 hours.
- The question only provides information about the speed for the 3rd hour (55 mph) which applies to the 5th hour as well.
- Therefore, the total distance traveled in 5 hours would be: Distance in hour 1 + Distance in hour 2 + Distance in hour 3 + Distance in hour 4 + Distance in hour 5 = 72 miles + 60 miles + 55 miles + (unknown) + 55 miles = 242 miles.
- However, the question asks for the total distance traveled in 5 hours, not the total possible distance considering the unknown speed in hour 4. So the answer is simply the sum of the known distances: 242 miles.
5. The Pizza Puzzle:
Answer: Make four straight cuts across the pizza, dividing it into 8 slices. Then, make one diagonal cut across any four slices that don't have olives, creating two larger slices. Your friend can take the four slices without olives, and you can take the remaining four slices (including two with olives and two without). This ensures both of you get an equal share (4 slices each) with and without olives.
