The Knave of Hearts stole some tarts. He ate half of the,, and half a tart more. The Knave of Diamonds ate half of what was left, and half a tart more. Then the Knave of Clubs ate half of what remained, and half a tart more. This left out just one tart for the Knave of Spades. How many tarts did the Knave of Hearts steal?
Let's work backwards to find out how many tarts the Knave of Hearts stole.
Given that there was only one tart left for the Knave of Spades, we can determine that the Knave of Clubs ate half a tart and half a tart more. So before the Knave of Clubs ate, there were (1 + 0.5) * 2 = 3 tarts remaining.
Now, before the Knave of Diamonds ate, there were (3 + 0.5) * 2 = 7 tarts remaining.
Before the Knave of Hearts ate, there were (7 + 0.5) * 2 = 15 tarts remaining.
Therefore, the Knave of Hearts stole a total of 15 tarts.
To solve this problem, we can work backwards from the final situation where only one tart is left for the Knave of Spades.
Let's go step by step:
1. The Knave of Spades had only one tart left.
2. Before this, the Knave of Clubs ate half of what remained and half a tart more.
So, he had (1 + 0.5) * 2 = 3 tarts.
3. Before this, the Knave of Diamonds ate half of what remained and half a tart more.
So, he had (3 + 0.5) * 2 = 7 tarts.
4. Before this, the Knave of Hearts ate half of what remained and half a tart more.
So, he had (7 + 0.5) * 2 = 15 tarts.
Therefore, the Knave of Hearts stole 15 tarts.
Let's break down the information step-by-step to find the answer:
1. The Knave of Hearts stole some tarts.
2. He ate half of the tarts and half a tart more.
3. The Knave of Diamonds ate half of what was left and half a tart more.
4. The Knave of Clubs ate half of what remained and half a tart more.
5. This left out just one tart for the Knave of Spades.
Let's start by figuring out how many tarts were remaining after each step:
Step 2: The Knave of Hearts ate half of the tarts and half a tart more.
Let's represent the initial number of tarts as "x":
- He ate half of the tarts: x/2
- He also ate half a tart more: 1/2
So, after the Knave of Hearts, there were x/2 - 1/2 tarts left.
Step 3: The Knave of Diamonds ate half of what was left and half a tart more.
Using the remaining tarts from Step 2 (x/2 - 1/2):
- He ate half of the remaining tarts: (x/2 - 1/2) * 1/2
- He also ate half a tart more: 1/2
So, after the Knave of Diamonds, there were (x/2 - 1/2) * 1/2 - 1/2 tarts left.
Step 4: The Knave of Clubs ate half of what remained and half a tart more.
Using the remaining tarts from Step 3 ((x/2 - 1/2) * 1/2 - 1/2):
- He ate half of the remaining tarts: ((x/2 - 1/2) * 1/2 - 1/2) * 1/2
- He also ate half a tart more: 1/2
So, after the Knave of Clubs, there were ((x/2 - 1/2) * 1/2 - 1/2) * 1/2 - 1/2 tarts left.
Step 5: This left out just one tart for the Knave of Spades.
Using the remaining tarts from Step 4 (((x/2 - 1/2) * 1/2 - 1/2) * 1/2 - 1/2):
- The remaining tarts are equal to 1.
Now, we can set up an equation and solve for "x" to find out how many tarts the Knave of Hearts stole:
1 = (((x/2 - 1/2) * 1/2 - 1/2) * 1/2 - 1/2)
Simplifying the equation:
1 = ((x/4 - 1/4) * 1/2 - 1/2) * 1/2 - 1/2
1 = (x/8 - 1/8 - 1/2) * 1/2 - 1/2
1 = (x/8 - 3/8) * 1/2 - 1/2
1 = (x/8 - 3/8)/2 - 1/2
1 = (x/8 - 3/8 - 4/8)/2
1 = (x/8 - 7/8)/2
2 = x/8 - 7/8
2 + 7/8 = x/8
23/8 = x/8
Multiplying both sides by 8 to isolate "x":
23 = x
Therefore, the Knave of Hearts stole 23 tarts.