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There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-
RULES:
- A person is allowed to enter the session if he/she can tell his/her own cap's colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in.
- No person is allowed to speak anything until he gets inside the session.
- The friends do not know the total varieties of colours of caps.
- The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their cap's colour and everyone is aware of this.
- All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds.
- The following events took place at the end of every minute.
EVENTS:
- 6 of the friends gained entry after the 1st time the gate was opened.
- The friends wearing blue and green caps entered the gate when it was opened for the second time.
- 4 persons entered the gate when it was opened for the third time.
- No one could enter the gate when it was opened for the fourth time.
- Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.
Q. How many different colours of caps were given to the 35 friends in total?
- a)8
- b)10
- c)9
- d)Cannot be determined
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
There were 35 logical friends who were trying to get tickets for a ses...
Analysis:
The key to solving this puzzle lies in analyzing the events that took place at the end of every minute. Let's break it down step by step:
1. Entry after the 1st time:
- 6 friends entered after the 1st time the gate was opened.
2. Entry after the 2nd time:
- Friends wearing blue and green caps entered when the gate was opened for the second time.
3. Entry after the 3rd time:
- 4 friends entered when the gate was opened for the third time.
4. No entry after the 4th time:
- No one could enter when the gate was opened for the fourth time.
5. Entry after the 5th time:
- Friends wearing multiple cap colors entered when the gate was opened for the fifth time.
Conclusion:
From the given events, we can deduce that there are 9 different colors of caps given to the 35 friends in total. This can be calculated by adding the friends who entered after each round: 6 (1st round) + 2 (2nd round) + 4 (3rd round) + 0 (4th round) + 1 (5th round) = 13. Since there are 35 friends in total, the remaining friends who did not enter in the first 5 rounds (35 - 13 = 22) must have different colored caps. Therefore, the total number of different colors of caps is 13 (friends who entered) + 9 (remaining friends) = 9.
Therefore, the correct answer is option C) 9.
The key to solving this puzzle lies in analyzing the events that took place at the end of every minute. Let's break it down step by step:
1. Entry after the 1st time:
- 6 friends entered after the 1st time the gate was opened.
2. Entry after the 2nd time:
- Friends wearing blue and green caps entered when the gate was opened for the second time.
3. Entry after the 3rd time:
- 4 friends entered when the gate was opened for the third time.
4. No entry after the 4th time:
- No one could enter when the gate was opened for the fourth time.
5. Entry after the 5th time:
- Friends wearing multiple cap colors entered when the gate was opened for the fifth time.
Conclusion:
From the given events, we can deduce that there are 9 different colors of caps given to the 35 friends in total. This can be calculated by adding the friends who entered after each round: 6 (1st round) + 2 (2nd round) + 4 (3rd round) + 0 (4th round) + 1 (5th round) = 13. Since there are 35 friends in total, the remaining friends who did not enter in the first 5 rounds (35 - 13 = 22) must have different colored caps. Therefore, the total number of different colors of caps is 13 (friends who entered) + 9 (remaining friends) = 9.
Therefore, the correct answer is option C) 9.
Community Answer
There were 35 logical friends who were trying to get tickets for a ses...
It is given that the 35 friends were able to figure out themselves the colour of the cap they were wearing. If this is the case, there should not be any cap colour present in a quantity of 1. Because then the person wearing it will never know the colour of his cap.
Therefore, it is necessary that the minimum number of caps of a particular colour is 2. Now, since the 35 friends are logical, the ones who see only 1 cap of a particular colour can understand that the colour of his cap is the same. It is given that after the gate was opened for the first time, 6 friends went inside. This suggests that 3 pairs of friends moved inside. So, 3 colours were present in numbers of 2.
Next up, out of the 29 friends left, everyone knows that the ones wearing 2 caps are gone. They cannot comment anything on their cap's colour watching 3 or more caps of the same colour. Because they cannot surely know if they are the ones with the fourth cap of the same colour. So, the ones who see only 2 caps of the same colour can understand that they are the ones wearing the cap with the same colour as the ones who were in pairs already left. The group of friends wearing blue and green cap entered the session when the gate was opened for the second time. So, 6 people joined the session and 23 were left outside
We now see a pattern building up. After people with 2 and 3 same coloured caps left, the ones with 4 same coloured caps will enter the gate and since 4 people left when the gate was open for the third time indicates that there was only 1 such colour which had 4 caps. Now, only 19 friends were outside the gate.
Again, we see that there was no group of 5 caps having the same colour. Had there been a group of 5 of the same coloured caps, they would have left the group.
The ones who could see that there are 5 other caps with same colour, understood that they are the ones wearing the 6th cap. But it is said that there were friends wearing multiple coloured caps who went inside for the session. So, either 2 or 3 groups of 6 people each must have left for the session. But, if there were 3 such groups, only 1 person would be left and we already saw that a single coloured cap is not possible. So, there were 2 such groups of caps which had 6 similar colour. So, 12 friends went inside leaving just 7 behind.
Now, each of the 7 fiends could see 6 caps of the same colour and thus conclude that they are the ones wearing the same coloured cap and go inside together.
So, in total, after the first round- 3 colours went inside.
After the second round, 2 colours went inside.
After the third round, 1 colour went inside.
After the 4th round- 0 colour went inside.
After the 5th round, 2 colours went inside and after the 6th round, 1 colour went inside.
∴ There were 9 colours of the cap with the 35 friends.
Therefore, it is necessary that the minimum number of caps of a particular colour is 2. Now, since the 35 friends are logical, the ones who see only 1 cap of a particular colour can understand that the colour of his cap is the same. It is given that after the gate was opened for the first time, 6 friends went inside. This suggests that 3 pairs of friends moved inside. So, 3 colours were present in numbers of 2.
Next up, out of the 29 friends left, everyone knows that the ones wearing 2 caps are gone. They cannot comment anything on their cap's colour watching 3 or more caps of the same colour. Because they cannot surely know if they are the ones with the fourth cap of the same colour. So, the ones who see only 2 caps of the same colour can understand that they are the ones wearing the cap with the same colour as the ones who were in pairs already left. The group of friends wearing blue and green cap entered the session when the gate was opened for the second time. So, 6 people joined the session and 23 were left outside
We now see a pattern building up. After people with 2 and 3 same coloured caps left, the ones with 4 same coloured caps will enter the gate and since 4 people left when the gate was open for the third time indicates that there was only 1 such colour which had 4 caps. Now, only 19 friends were outside the gate.
Again, we see that there was no group of 5 caps having the same colour. Had there been a group of 5 of the same coloured caps, they would have left the group.
The ones who could see that there are 5 other caps with same colour, understood that they are the ones wearing the 6th cap. But it is said that there were friends wearing multiple coloured caps who went inside for the session. So, either 2 or 3 groups of 6 people each must have left for the session. But, if there were 3 such groups, only 1 person would be left and we already saw that a single coloured cap is not possible. So, there were 2 such groups of caps which had 6 similar colour. So, 12 friends went inside leaving just 7 behind.
Now, each of the 7 fiends could see 6 caps of the same colour and thus conclude that they are the ones wearing the same coloured cap and go inside together.
So, in total, after the first round- 3 colours went inside.
After the second round, 2 colours went inside.
After the third round, 1 colour went inside.
After the 4th round- 0 colour went inside.
After the 5th round, 2 colours went inside and after the 6th round, 1 colour went inside.
∴ There were 9 colours of the cap with the 35 friends.
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Question Description
There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? for CAT 2026 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?.
There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? for CAT 2026 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?.
Solutions for There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT. Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice There were 35 logical friends who were trying to get tickets for a session held by a famous Mathematician- Mr X. But Mr X put in a challenge for all 35 of the friends to get into the session. He gave a cap each to the 35 enthusiastic friends, the colour of which could not be seen by the person wearing it. However, a person was able to see the colours of the caps worn by the remaining 34 friends. Given below are some rules put in by Mr X for the 35 friends-RULES: A person is allowed to enter the session if he/she can tell his/her own caps colour. Mr X gives 1 minute time for them to think, at the end of which he opens the door. He repeats this process till everyone gets in. No person is allowed to speak anything until he gets inside the session. The friends do not know the total varieties of colours of caps. The caps are distributed in such a way that it will eventually be possible for everyone to logically deduce their caps colour and everyone is aware of this. All the 35 friends played the game perfectly and gained the entry eventually after multiple rounds. The following events took place at the end of every minute.EVENTS: 6 of the friends gained entry after the 1st time the gate was opened. The friends wearing blue and green caps entered the gate when it was opened for the second time. 4 persons entered the gate when it was opened for the third time. No one could enter the gate when it was opened for the fourth time. Friends wearing multiple cap colours entered the gate when it was opened for the fifth time.Q.How many different colours of caps were given to the 35 friends in total?a)8b)10c)9d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice CAT tests.
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