answer:This is a classic logic puzzle involving common knowledge and induction. To free the prisoners without breaking the rules—namely, that your statement cannot convey new information—you can make the following announcement to all the prisoners: "I see at least one person on this island with green eyes." This statement does not convey any new information, since each prisoner can already see at least one other prisoner with green eyes. However, it introduces common knowledge that there is at least one green-eyed person, which is information all prisoners already had, but now they all know that everyone is aware of it. What happens next is a process of logical deduction that unfolds over 100 days: 1. If there were only one green-eyed prisoner, upon hearing your announcement, that prisoner would realize that the "at least one person" must be them because they cannot see anyone else with green eyes. Thus, they would leave the island at midnight on the first day. 2. Since no one leaves on the first night, it must mean there are at least two green-eyed prisoners. Each green-eyed prisoner can deduce this because they can see at least one person with green eyes, and they know that person did not leave because they also heard there was at least one other green-eyed person. 3. If there were only two green-eyed prisoners, on the second night each would expect the other to have left, realizing the announcement would have prompted their escape if they were the only one. When both are still there on the second day, each can deduce there must be at least three green-eyed prisoners. This pattern continues each day. When no prisoners leave by the nth night, each prisoner realizes that if there were only n green-eyed prisoners, each would have expected the others to leave on the nth night. Their continued presence implies there are at least n+1 green-eyed prisoners. After 99 days, each of the 100 prisoners can deduce that if there were only 99 green-eyed prisoners, they would have all left on the 99th night. Since this does not happen, on the 100th day, each green-eyed prisoner realizes that they themselves must have green eyes too, thus counting 100 green-eyed prisoners. They all go to the guards at midnight on the 100th day, declare they have green eyes, and are set free according to the rules.

answer:The farmer can follow these steps to get the fox, chicken, and sack of grain across the river without any of them being eaten: 1. First, the farmer takes the chicken across the river and leaves it on the other side. He leaves the fox and the grain together on the starting side, as the fox won't eat the grain. Starting side: Fox, Grain Boat: Farmer, Chicken Other side: 2. The farmer then goes back alone to the original side to get the fox. Starting side: Grain Boat: Farmer Other side: Chicken 3. The farmer takes the fox across the river, but he can't leave the fox and chicken together. So he takes the chicken back on the boat when he returns to the original side to pick up the grain. Starting side: Chicken, Grain Boat: Farmer, Fox Other side: 4. The farmer leaves the fox on the other side and goes back with the boat to get the chicken. Starting side: Chicken Boat: Farmer Other side: Fox 5. Lastly, the farmer takes the grain across, leaving the chicken on the starting side. Starting side: Chicken Boat: Farmer, Grain Other side: Fox 6. The farmer leaves the grain with the fox on the other side – as they won't harm each other – and goes back to get the chicken. Starting side: Boat: Farmer, Chicken Other side: Fox, Grain 7. Finally, the farmer takes the chicken across the river again. Now, all three - the fox, the chicken, and the sack of grain - are safely on the other side. Starting side: Boat: Farmer Other side: Fox, Chicken, Grain The farmer has successfully taken the fox, the chicken, and the sack of grain across the river without anything being eaten.

answer:To boil an egg in exactly 15 minutes using a 7-minute hourglass and an 11-minute hourglass, you can follow these steps: 1. Start by flipping both hourglasses over at the same time and begin boiling the egg. 2. When the 7-minute hourglass runs out, 7 minutes have passed. Keep the 11-minute hourglass running, and flip the 7-minute hourglass over immediately to start timing another 7-minute interval. 3. After the 11-minute hourglass runs out (4 minutes after the 7-minute one does), 11 minutes in total have passed. Since the 7-minute hourglass has been flipped, 4 minutes of sand have dropped from the top to the bottom. There are now 3 minutes left until the 7-minute hourglass runs out again. 4. As soon as the 11-minute hourglass is empty, flip it over to start timing another 11 minutes, but in this case, we will only be interested in the first 4 minutes. 5. When the 7-minute hourglass runs out a second time, exactly 14 minutes have elapsed (7 minutes from the first run and 7 minutes from the second). 6. Since you flipped the 11-minute hourglass again when there were exactly 3 minutes left on the 7-minute hourglass, there should now be 3 minutes of sand at the top of the 11-minute hourglass, and 1 minute's worth of sand will have fallen to the bottom. 7. Flip the 7-minute hourglass one more time as soon as it runs out the second time. At this point, allow the sand to fall for exactly 1 more minute in the 11-minute hourglass until the sand reaches the top of the hourglass section. This will mark exactly 15 minutes. 8. At 15 minutes, turn off the heat and remove the egg. It's been cooking for exactly 15 minutes. To recap the steps and orientations: - At 0 minutes: Start both hourglasses and the egg. - At 7 minutes: 7-minute hourglass runs out; flip 7-minute hourglass. - At 11 minutes: 11-minute hourglass runs out; flip 11-minute hourglass. - At 14 minutes: 7-minute hourglass runs out for the second time; flip 7-minute hourglass. - At 15 minutes: 1 minute has passed on the 11-minute hourglass; the egg is done.

answer:To determine which road to take, you need to devise a question such that, regardless of which brother you ask (the liar or the truth-teller), you get a reliable answer about the correct path. The key is to ask a question that will force both brothers to point to the same road, the one that leads to eternal happiness. You can ask either brother the following question: "If I were to ask your brother which road leads to eternal happiness, what would he say?" Let's analyze the possible scenarios: 1. If you ask the truth-telling brother, he will tell you truthfully what his lying brother would say. Since the lying brother always lies, he would point you to the road that leads to certain death. Knowing this, the truth-teller will tell you that his brother would point to the road that leads to certain death. 2. If you ask the lying brother, he will lie about the answer his truth-telling brother would give. The truth-telling brother would point you to the road that leads to eternal happiness, but since the lying brother never tells the truth, he would tell you that his brother would point to the road that leads to certain death. In both scenarios, regardless of which brother you ask, the answer will point to the road that leads to certain death since: - The truth-teller would honestly tell you the lie that the liar would give. - The liar would falsely tell you the true response that the truth-teller would give. So, given the responses, you would choose the opposite of the road they indicate, which will be the road that leads to eternal happiness.