A commander wishes to draw up his 8500 soldiers in the form of a square. After arranging, he found out that some of them are left out. How many soldiers were left out?
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Answers
Answer:
Hint: Firstly, we will apply approximation and find the number whose square is closest to the number of total soldiers which is 7500. Then, we will subtract the squared number value from the total number of the soldiers. Then, on subtraction we get the number of the soldiers who are left out to form a largest square.
Complete step-by-step answer:
In this question, we have to go by estimation as it requires a number which is a perfect square closer to the value 7500.
Here, the number of the soldiers is 7500 and they need to be arranged in the form of the largest square.
It is clearly evident by seeing that it is less than the square of 90 as the square of 90 is:
902=8100
Now, by applying approximation in the question and choosing the number smaller than 90 to get its perfect square.
So, we choose the number as 85 and find its square as:
852=7225
The square of 85 is less than 7500 then we should move to the next number which is 86.
Find the square of 86 as:
862=7396
Again, the square of 86 is smaller than 7500 then we should move to the next number which is 87
Find the square of 87 as:
872=7569
The square of the number 87 gives the value larger than 7500. So, it is clear that we need to subtract the square of the number 86 from the value 7500 to get the remaining soldiers to form a largest square.
Now, to get the number of the soldiers left is:
7500−7396=104
Hence, 104 soldiers are left out.
Note: Another approach to solve this type of the problem is:
Firstly we can find the square root of the total number of the soldiers given which automatically gives us the closest approximation of the nearest natural number whose complete square is possible.
So, by finding the square root of 7500 as:
7500−−−−√=86.6
It is clear from the above calculation that the number which makes the largest square from the given number of soldiers is 86.
Now, to get the number of the soldiers left is:
7500−862=7500−7396⇒104
Hence, 104 soldiers are left out.
Step-by-step explanation:
The grandfather paradox is a potential logical problem that would arise if a person were to travel to a past time. The name comes from the idea that if a person travels to a time before their grandfather had children, and kills him, it would make their own birth impossible. So, if time travel is possible, it somehow must avoid such a contradiction.
The logical inconsistency of time travel is a common theme in time-warping fiction, but it's also of interest to philosophers. In early versions of the grandfather paradox, some tried to argue that time travel was impossible on logical grounds, said Tim Maudlin, a philosopher at New York University, who frequently writes about physics and philosophy. "In a way, that's like asking why, right now, I can't be wet and completely dry," he said. "Well, that's just logically impossible. What are you asking about?"
But contradictions such as the grandfather paradox don't mean that time travel is impossible. The logical consistency of time travel largely depends on the concept of time, and physicists have many different ways of conceptualizing time. For example, if some laws of physics are considered probabilistic, rather than precisely determined, it opens the possibility of multiple outcomes from a trip back in time, some of which may not be contradictory.
"It's actually more difficult than you think to come up with a situation where there are no consistent solutions," Maudlin said. For a logically consistent time travel story, he gives the example of a traveler going back in time to shoot themselves. They aim to kill, but miss because of a tremor in their hand. The non-lethal shot strikes a nerve in the past version of the traveler, causing a tremor in their hand for the rest of their life.
The concept of time travel can also be separated from the idea of changing the past, or backward causation. But Maudlin doesn't think backward causation is possible. "I think it's contrary to the nature of time itself," he said. "That's not a majority view."