💡Okay, so I've got this probability problem here: "A fair die is rolled eight times. What is the probability of rolling at least a four at least seven times?" The options are A) 1/32, B) 9/256, C) 1/256, and D) 8/256. Hmm, let me try to figure this out step by step.First, I need to understand what's being asked. Rolling a die eight times, and we want the probability that we get at least a four (which means rolling a 4, 5, or 6) at least seven times. So, that could mean getting exactly seven fours, fives, or sixes, or getting exactly eight of them. So, I think I need to calculate the probability for both scenarios and then add them together.Alright, let's break it down. When rolling a die, the probability of getting at least a four on a single roll is... well, there are three favorable outcomes (4, 5, 6) out of six possible outcomes. So, that's 3/6, which simplifies to 1/2. So, the probability of success (rolling a 4, 5, or 6) is 1/2, and the probability of failure (rolling a 1, 2, or 3) is also 1/2.Since each roll is independent, I can use the binomial probability formula here. The binomial formula is P(k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the combination of n things taken k at a time.So, for exactly seven successes (i.e., rolling at least a four seven times), n is 8, k is 7, p is 1/2. Let's compute that.First, C(8,7) is the number of ways to choose 7 successes out of 8 trials. That's 8, because there are 8 different rolls where the failure could occur. So, C(8,7) = 8.Then, p^7 is (1/2)^7, which is 1/128. And (1-p)^(8-7) is (1/2)^1, which is 1/2. So, multiplying these together: 8 * (1/128) * (1/2) = 8 * (1/256) = 8/256. Simplifying that, 8 divided by 256 is 1/32. So, the probability of exactly seven successes is 1/32.Now, let's compute the probability of exactly eight successes, which means rolling at least a four all eight times. Here, k is 8, so C(8,8) is 1, since there's only one way to have all eight rolls be successes.p^8 is (1/2)^8, which is 1/256. And (1-p)^(8-8) is (1/2)^0, which is 1. So, the probability is 1 * (1/256) * 1 = 1/256.Now, to find the total probability of getting at least seven successes, we add the probabilities of exactly seven and exactly eight successes. So, that's 1/32 + 1/256.To add these fractions, they need a common denominator. 1/32 is equivalent to 8/256. So, 8/256 + 1/256 = 9/256.Looking back at the options, B is 9/256, so that should be the correct answer.Wait, let me double-check my calculations to make sure I didn't make a mistake.For exactly seven successes: C(8,7) = 8, (1/2)^7 = 1/128, (1/2)^1 = 1/2. So, 8 * (1/128) * (1/2) = 8/256 = 1/32. That seems right.For exactly eight successes: C(8,8) = 1, (1/2)^8 = 1/256. So, 1 * 1/256 = 1/256. Adding them together: 1/32 + 1/256 = 8/256 + 1/256 = 9/256. Yep, that checks out.I think I got it right. So, the answer should be B) 9/256.

💡Alright, so I have this problem where I need to find the position of ( x_{21} ) in ( P_3 ) when ( N = 32 ). Let me try to understand the process step by step.First, the problem describes something called a "C transformation." From what I gather, a C transformation involves taking a permutation of numbers and rearranging them by separating the numbers in odd positions from those in even positions. Then, you place all the odd-positioned numbers first, followed by the even-positioned ones. This seems similar to a perfect shuffle or something like that.Given ( N = 32 ), which is ( 2^5 ), we start with ( P_0 ), which is just the sequence ( x_1, x_2, x_3, ldots, x_{32} ). Applying the C transformation to ( P_0 ) gives ( P_1 ). So, ( P_1 ) should be ( x_1, x_3, x_5, ldots, x_{31}, x_2, x_4, x_6, ldots, x_{32} ). That makes sense because we're taking all the odd-indexed elements first, then the even-indexed ones.Next, to get ( P_2 ), we need to divide ( P_1 ) into two segments, each containing ( frac{N}{2} = 16 ) numbers, and apply the C transformation to each segment. So, the first segment is ( x_1, x_3, x_5, ldots, x_{31} ) and the second segment is ( x_2, x_4, x_6, ldots, x_{32} ). Applying the C transformation to each segment would mean separating the odd and even positions within each segment.Let me try to write out ( P_2 ):- First segment after C transformation: ( x_1, x_5, x_9, x_{13}, x_{17}, x_{21}, x_{25}, x_{29}, x_3, x_7, x_{11}, x_{15}, x_{19}, x_{23}, x_{27}, x_{31} )- Second segment after C transformation: ( x_2, x_6, x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{30}, x_4, x_8, x_{12}, x_{16}, x_{20}, x_{24}, x_{28}, x_{32} )So, putting them together, ( P_2 ) is:( x_1, x_5, x_9, x_{13}, x_{17}, x_{21}, x_{25}, x_{29}, x_3, x_7, x_{11}, x_{15}, x_{19}, x_{23}, x_{27}, x_{31}, x_2, x_6, x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{30}, x_4, x_8, x_{12}, x_{16}, x_{20}, x_{24}, x_{28}, x_{32} )Now, to get ( P_3 ), we need to divide ( P_2 ) into four segments, each containing ( frac{N}{4} = 8 ) numbers, and apply the C transformation to each segment.Let me identify the four segments in ( P_2 ):1. First segment: ( x_1, x_5, x_9, x_{13}, x_{17}, x_{21}, x_{25}, x_{29} )2. Second segment: ( x_3, x_7, x_{11}, x_{15}, x_{19}, x_{23}, x_{27}, x_{31} )3. Third segment: ( x_2, x_6, x_{10}, x_{14}, x_{18}, x_{22}, x_{26}, x_{30} )4. Fourth segment: ( x_4, x_8, x_{12}, x_{16}, x_{20}, x_{24}, x_{28}, x_{32} )Now, applying the C transformation to each segment:1. First segment after C transformation: ( x_1, x_9, x_{17}, x_{25}, x_5, x_{13}, x_{21}, x_{29} )2. Second segment after C transformation: ( x_3, x_{11}, x_{19}, x_{27}, x_7, x_{15}, x_{23}, x_{31} )3. Third segment after C transformation: ( x_2, x_{10}, x_{18}, x_{26}, x_6, x_{14}, x_{22}, x_{30} )4. Fourth segment after C transformation: ( x_4, x_{12}, x_{20}, x_{28}, x_8, x_{16}, x_{24}, x_{32} )Putting all these transformed segments together, ( P_3 ) becomes:( x_1, x_9, x_{17}, x_{25}, x_5, x_{13}, x_{21}, x_{29}, x_3, x_{11}, x_{19}, x_{27}, x_7, x_{15}, x_{23}, x_{31}, x_2, x_{10}, x_{18}, x_{26}, x_6, x_{14}, x_{22}, x_{30}, x_4, x_{12}, x_{20}, x_{28}, x_8, x_{16}, x_{24}, x_{32} )Looking at this permutation, I need to find where ( x_{21} ) is located. Scanning through the list:1. ( x_1 )2. ( x_9 )3. ( x_{17} )4. ( x_{25} )5. ( x_5 )6. ( x_{13} )7. ( x_{21} )8. ( x_{29} )9. ( x_3 )10. ( x_{11} )11. ( x_{19} )12. ( x_{27} )13. ( x_7 )14. ( x_{15} )15. ( x_{23} )16. ( x_{31} )17. ( x_2 )18. ( x_{10} )19. ( x_{18} )20. ( x_{26} )21. ( x_6 )22. ( x_{14} )23. ( x_{22} )24. ( x_{30} )25. ( x_4 )26. ( x_{12} )27. ( x_{20} )28. ( x_{28} )29. ( x_8 )30. ( x_{16} )31. ( x_{24} )32. ( x_{32} )So, ( x_{21} ) is at position 7 in ( P_3 ). That seems straightforward, but let me double-check to make sure I didn't make a mistake in the transformations.Starting from ( P_0 ), applying C transformation to get ( P_1 ), then dividing into two segments and applying C to each to get ( P_2 ), and then dividing into four segments and applying C to each to get ( P_3 ). Each step seems to follow the rules described. The position of ( x_{21} ) in ( P_3 ) is indeed 7.I think that's correct. Maybe I can think of another way to verify this, perhaps by considering the binary representation of the positions or something related to bit-reversal, since these transformations often relate to binary shifts or reversals. But since I'm getting position 7 through the step-by-step process, and it seems consistent, I'll go with that.**Final Answer**The position of ( x_{21} ) in ( P_3 ) is boxed{7}.

💡First, I'll start by expressing all the symbols in terms of one variable to simplify the equations.Given that (triangle = bigcirc + 2), I can express (bigcirc) as (triangle - 2).Next, since (square = triangle + triangle), this simplifies to (square = 2triangle).For the star symbol, (star = triangle + square + 5). Substituting the expression for (square) from the previous step, we get (star = triangle + 2triangle + 5 = 3triangle + 5).We are also given that (star = bigcirc + 31). Substituting the expression for (bigcirc) from the first step, this becomes (3triangle + 5 = (triangle - 2) + 31).Simplifying this equation:[3triangle + 5 = triangle + 29]Subtracting (triangle) and 5 from both sides:[2triangle = 24]Dividing both sides by 2:[triangle = 12]Now that we have the value of (triangle), we can find the other symbols:[bigcirc = triangle - 2 = 12 - 2 = 10][square = 2triangle = 2 times 12 = 24][star = 3triangle + 5 = 3 times 12 + 5 = 41]

💡Okay, so I have this problem where for a natural number ( n ), ( f(n) ) is defined as the number of positive integers less than ( n ) which are neither coprime to ( n ) nor a divisor of it. I need to prove that for each positive integer ( k ), there are only finitely many ( n ) satisfying ( f(n) = k ).Hmm, let me try to break this down. First, I should understand what ( f(n) ) represents. It counts numbers less than ( n ) that aren't coprime to ( n ) and also aren't divisors of ( n ). So, if I think about the numbers from 1 to ( n-1 ), some are coprime to ( n ), some are divisors, and some are neither. ( f(n) ) is the count of those neither.I remember that Euler's totient function ( varphi(n) ) counts the numbers less than ( n ) that are coprime to ( n ). Also, the number of divisors of ( n ) is denoted by ( d(n) ). So, maybe I can express ( f(n) ) in terms of ( varphi(n) ) and ( d(n) ).Let's see. The total number of positive integers less than ( n ) is ( n - 1 ). Out of these, ( varphi(n) ) are coprime to ( n ), and ( d(n) - 1 ) are the proper divisors (since ( n ) itself isn't counted). But wait, some numbers might be both coprime and divisors? No, actually, if a number is a divisor, it can't be coprime unless it's 1. So, 1 is the only number that is both a divisor and coprime to ( n ). So, the count of numbers that are either coprime or divisors is ( varphi(n) + d(n) - 1 ). Therefore, the numbers that are neither would be ( (n - 1) - (varphi(n) + d(n) - 1) = n - varphi(n) - d(n) ).So, ( f(n) = n - varphi(n) - d(n) ). Got it. Now, I need to show that for each ( k ), there are only finitely many ( n ) such that ( f(n) = k ).Hmm, so I need to show that as ( n ) grows, ( f(n) ) grows without bound, meaning it can't stay equal to any fixed ( k ) beyond some point. So, if I can show that ( f(n) ) tends to infinity as ( n ) tends to infinity, then for any fixed ( k ), there can only be finitely many ( n ) with ( f(n) = k ).But wait, is ( f(n) ) always increasing? Not necessarily, because ( varphi(n) ) and ( d(n) ) can behave differently. For example, for prime numbers, ( varphi(p) = p - 1 ) and ( d(p) = 2 ), so ( f(p) = p - (p - 1) - 2 = -1 ), but since ( f(n) ) is defined as a count, maybe it's 0 for primes? Wait, let me check.If ( n ) is prime, then all numbers less than ( n ) are coprime to ( n ), except 1, which is a divisor. So, the numbers less than ( n ) are either coprime or divisors. Therefore, there are no numbers that are neither, so ( f(n) = 0 ) for primes. So, ( f(n) ) is zero for primes, which are infinitely many. But the problem says for each positive integer ( k ), there are only finitely many ( n ) with ( f(n) = k ). So, ( k = 0 ) is a special case, but the problem specifies positive integers ( k ), so ( k geq 1 ).So, for ( k geq 1 ), I need to show that only finitely many composite numbers ( n ) satisfy ( f(n) = k ).Alright, so let's focus on composite numbers. For composite ( n ), ( f(n) = n - varphi(n) - d(n) ). I need to analyze how ( f(n) ) behaves as ( n ) grows.I know that ( varphi(n) ) is approximately ( n prod_{p|n} left(1 - frac{1}{p}right) ). So, for numbers with many small prime factors, ( varphi(n) ) is smaller, meaning ( n - varphi(n) ) is larger. On the other hand, ( d(n) ) is the number of divisors, which tends to be larger for numbers with many prime factors.So, ( f(n) ) is the difference between ( n - varphi(n) ) and ( d(n) ). If I can show that ( n - varphi(n) ) grows faster than ( d(n) ), then ( f(n) ) would tend to infinity, implying that only finitely many ( n ) can have ( f(n) = k ) for any fixed ( k ).Let me think about how ( n - varphi(n) ) behaves. For a prime power ( p^k ), ( varphi(p^k) = p^k - p^{k-1} ), so ( n - varphi(n) = p^{k-1} ). For numbers with multiple prime factors, ( n - varphi(n) ) is larger. For example, for ( n = pq ) where ( p ) and ( q ) are distinct primes, ( varphi(n) = (p - 1)(q - 1) ), so ( n - varphi(n) = pq - (p - 1)(q - 1) = p + q - 1 ). So, it's linear in the primes.On the other hand, ( d(n) ) for ( n = pq ) is 4, which is constant. So, ( f(n) = (p + q - 1) - 4 ). As ( p ) and ( q ) grow, ( f(n) ) grows without bound. So, for semiprimes, ( f(n) ) can be made arbitrarily large by choosing large primes.Similarly, for numbers with more prime factors, ( n - varphi(n) ) increases, while ( d(n) ) increases but perhaps not as fast. For example, take ( n = 2^k ). Then ( varphi(n) = 2^k - 2^{k-1} = 2^{k-1} ), so ( n - varphi(n) = 2^{k-1} ). ( d(n) = k + 1 ). So, ( f(n) = 2^{k-1} - (k + 1) ). As ( k ) increases, ( 2^{k-1} ) grows exponentially, while ( k + 1 ) grows linearly, so ( f(n) ) tends to infinity.Another example: ( n = p_1 p_2 dots p_m ), the product of the first ( m ) primes. Then ( varphi(n) = n prod_{i=1}^m left(1 - frac{1}{p_i}right) ). So, ( n - varphi(n) = n left(1 - prod_{i=1}^m left(1 - frac{1}{p_i}right)right) ). This is known to grow roughly like ( n cdot frac{1}{log log n} ), which is still quite large. Meanwhile, ( d(n) = 2^m ), which is exponential in ( m ), but ( n ) is the product of the first ( m ) primes, which grows faster than exponentially. So, ( n - varphi(n) ) is much larger than ( d(n) ), so ( f(n) ) tends to infinity.So, in general, for composite ( n ), ( n - varphi(n) ) tends to be significantly larger than ( d(n) ), especially as ( n ) becomes large. Therefore, ( f(n) = n - varphi(n) - d(n) ) tends to infinity as ( n ) tends to infinity.Therefore, for any fixed positive integer ( k ), there can only be finitely many ( n ) such that ( f(n) = k ). Because beyond some point, ( f(n) ) will exceed any fixed ( k ) and keep growing.Wait, but I should make this more rigorous. Maybe I can use some inequalities or known results about ( varphi(n) ) and ( d(n) ).I recall that ( varphi(n) ) is at least ( frac{n}{2} ) for ( n geq 2 ), but actually, that's not true. For example, ( varphi(2^k) = 2^{k-1} ), which is ( frac{n}{2} ). For numbers with many small factors, ( varphi(n) ) can be much smaller. For example, ( varphi(n) ) can be as small as ( n prod_{p|n} left(1 - frac{1}{p}right) ). So, if ( n ) has many small prime factors, ( varphi(n) ) is small, making ( n - varphi(n) ) large.Also, I know that ( d(n) ) is ( O(n^{epsilon}) ) for any ( epsilon > 0 ). That is, the number of divisors grows slower than any positive power of ( n ). So, ( d(n) ) is much smaller than ( n ) for large ( n ).Therefore, ( f(n) = n - varphi(n) - d(n) ) is approximately ( n - varphi(n) ), since ( d(n) ) is negligible compared to ( n ). And ( n - varphi(n) ) is known to be at least ( frac{n}{2} ) for even ( n ), but actually, it can be much larger.Wait, let me think again. For ( n ) being a prime power, ( n - varphi(n) ) is ( p^{k-1} ), which is linear in ( p ) for fixed ( k ). But for numbers with many prime factors, ( n - varphi(n) ) can be much larger. For example, if ( n ) is the product of the first ( m ) primes, ( n - varphi(n) ) is roughly ( n cdot frac{1}{log log n} ), which is still a significant portion of ( n ).So, combining these, ( f(n) ) is roughly ( n - varphi(n) ), which tends to infinity as ( n ) grows, because ( varphi(n) ) is at most ( n - 1 ) (for primes, but primes have ( f(n) = 0 )), but for composites, ( varphi(n) ) is significantly smaller, so ( n - varphi(n) ) is large.Therefore, ( f(n) ) tends to infinity as ( n ) tends to infinity, meaning that for any fixed ( k ), there are only finitely many ( n ) with ( f(n) = k ).I think this makes sense. To make it more formal, I could use the fact that ( varphi(n) ) is multiplicative and for composite ( n ), ( varphi(n) leq n - sqrt{n} ) (I think this is a known result). If that's the case, then ( f(n) = n - varphi(n) - d(n) geq n - (n - sqrt{n}) - d(n) = sqrt{n} - d(n) ). Since ( d(n) ) is much smaller than ( sqrt{n} ) for large ( n ), ( f(n) ) is at least roughly ( sqrt{n} ), which tends to infinity. Therefore, ( f(n) ) must tend to infinity, implying only finitely many ( n ) for each ( k ).Yeah, that seems like a solid argument. So, to summarize, ( f(n) ) is equal to ( n - varphi(n) - d(n) ). For composite ( n ), ( varphi(n) ) is significantly smaller than ( n ), so ( n - varphi(n) ) is large. The number of divisors ( d(n) ) is much smaller than ( n ), so ( f(n) ) is dominated by ( n - varphi(n) ), which tends to infinity. Therefore, ( f(n) ) must tend to infinity as ( n ) grows, meaning only finitely many ( n ) can have ( f(n) = k ) for any fixed ( k ).I think that covers it. Maybe I should also consider specific cases or use known inequalities to make it more rigorous, but the general idea is that ( f(n) ) grows without bound, so each ( k ) can only be achieved finitely many times.

💡Okay, so I'm trying to figure out for which values of ( n ) it's possible to cover the surface of an ( n times n times n ) checkered cube with ( 1 times 2 ) paper rectangles in such a way that each rectangle borders exactly five other rectangles along its edges. Hmm, that sounds a bit tricky, but let me break it down step by step.First, I need to visualize the cube. An ( n times n times n ) cube has six faces, each face being an ( n times n ) grid of squares. So, the total number of squares on the surface of the cube is ( 6n^2 ). Since each ( 1 times 2 ) rectangle covers two squares, the total number of rectangles needed to cover the entire surface would be ( frac{6n^2}{2} = 3n^2 ).Now, the problem specifies that each rectangle must border exactly five other rectangles along its edges. Let me think about what that means. Each ( 1 times 2 ) rectangle has four edges, but since it's on the surface of a cube, some of these edges will be on the boundary of the cube and won't border another rectangle. Wait, no, actually, the rectangles are only covering the surface, so all edges of the rectangles are either on the boundary of the cube or adjacent to another rectangle.But the problem says each rectangle borders exactly five others. So, each rectangle must have five edges that are adjacent to other rectangles. But a ( 1 times 2 ) rectangle only has four edges. Hmm, that seems impossible because you can't have five adjacent edges on a rectangle that only has four edges. Wait, maybe I'm misunderstanding something.Oh, perhaps it's not the number of edges but the number of adjacent rectangles. Each rectangle has four edges, and each edge can border another rectangle. So, if each rectangle borders five others, that would mean that each rectangle has five adjacent rectangles, but since it only has four edges, one of those adjacencies must be along a corner or something? That doesn't make sense because adjacency along a corner doesn't count as bordering along an edge.Wait, maybe the problem is considering that each rectangle can be adjacent to other rectangles on multiple edges. For example, a rectangle could be adjacent to two rectangles along one of its longer edges and two along another, but that still only gives four adjacencies. Hmm, I'm confused.Let me think differently. Maybe it's about the number of neighboring rectangles, not the number of edges. So, each rectangle has four edges, and each edge can border at most one other rectangle. Therefore, the maximum number of adjacent rectangles a single rectangle can have is four. But the problem says five, which seems impossible. Maybe I'm misinterpreting the problem.Wait, perhaps the cube is being considered in 3D, so a rectangle on one face can border rectangles on adjacent faces. That might allow for more adjacencies. Let me consider that.Imagine a rectangle on the top face of the cube. It can border rectangles on the top face, and also potentially on the front, back, left, or right faces, depending on its position. So, if a rectangle is on an edge or corner of the cube, it might border rectangles on multiple faces. That could potentially increase the number of adjacent rectangles beyond four.But how exactly? Let me try to visualize a specific case. Let's say ( n = 2 ). A ( 2 times 2 times 2 ) cube has each face divided into four squares. Covering each face with ( 1 times 2 ) rectangles would require two rectangles per face, so a total of twelve rectangles for the entire cube.Now, each rectangle on a face is adjacent to other rectangles on the same face and potentially on adjacent faces. For example, a rectangle on the top face could border rectangles on the front, right, back, and left faces if it's positioned at the corner. But wait, each rectangle only has four edges, so even if it's at a corner, it can only border four other rectangles, right?Hmm, maybe not. If a rectangle is placed such that it spans across an edge of the cube, it might be adjacent to rectangles on two different faces. But in a ( 2 times 2 times 2 ) cube, each rectangle is confined to a single face because the cube is too small to have overlapping faces. So, in that case, each rectangle can only border other rectangles on the same face.Wait, but in a ( 2 times 2 times 2 ) cube, each face has only two rectangles, so each rectangle on a face can only border one other rectangle on that face. But the problem requires each rectangle to border five others, which is impossible in this case. So, maybe ( n = 2 ) is not possible?But earlier, I thought ( n = 2 ) might work. Maybe I need to reconsider. Perhaps the tiling isn't just on the faces but also wraps around the edges or something. But no, the problem specifies covering the surface, so it's just the outer faces.Wait, maybe I'm overcomplicating this. Let's think about the total number of adjacencies. Each rectangle borders five others, so the total number of adjacencies is ( 5 times 3n^2 ). But each adjacency is shared between two rectangles, so the total number of adjacencies should also be equal to twice the number of edges between rectangles.But each ( 1 times 2 ) rectangle has four edges, and each edge can be adjacent to another rectangle or not. So, the total number of potential adjacencies is ( 4 times 3n^2 ). But since each adjacency is counted twice, the actual number of adjacencies is ( 2 times 3n^2 ). Wait, that doesn't make sense.Let me clarify. The total number of edges on all rectangles is ( 4 times 3n^2 ). However, each adjacency between two rectangles corresponds to two edges (one from each rectangle). Therefore, the total number of adjacencies is ( frac{4 times 3n^2 - B}{2} ), where ( B ) is the number of edges on the boundary of the cube that are not adjacent to any other rectangle.But the problem states that each rectangle borders exactly five others, so the total number of adjacencies is ( 5 times 3n^2 ). But this must equal ( frac{4 times 3n^2 - B}{2} ). Therefore:[5 times 3n^2 = frac{4 times 3n^2 - B}{2}]Simplifying:[15n^2 = frac{12n^2 - B}{2}][30n^2 = 12n^2 - B][B = 12n^2 - 30n^2][B = -18n^2]Wait, that can't be right because ( B ) represents the number of boundary edges, which can't be negative. This suggests that my initial assumption might be flawed. Maybe the way I'm counting adjacencies is incorrect.Let me try a different approach. Each rectangle has four edges, and each edge can either border another rectangle or be on the boundary of the cube. Let ( E ) be the total number of edges on all rectangles, which is ( 4 times 3n^2 = 12n^2 ). Let ( B ) be the number of boundary edges (edges not adjacent to any other rectangle). Then, the number of adjacencies is ( frac{E - B}{2} ), since each adjacency is shared between two rectangles.Given that each rectangle borders exactly five others, the total number of adjacencies is ( 5 times 3n^2 = 15n^2 ). Therefore:[frac{12n^2 - B}{2} = 15n^2][12n^2 - B = 30n^2][-B = 18n^2][B = -18n^2]Again, I'm getting a negative number for ( B ), which doesn't make sense. This suggests that it's impossible for each rectangle to border exactly five others because it would require a negative number of boundary edges, which isn't possible.Wait, maybe I made a mistake in calculating the total number of adjacencies. If each rectangle borders five others, then the total number of adjacencies is indeed ( 5 times 3n^2 ). But each adjacency is counted twice (once for each rectangle), so the actual number of unique adjacencies is ( frac{5 times 3n^2}{2} ). Let me adjust that.So, the total number of unique adjacencies is ( frac{15n^2}{2} ). But the number of unique adjacencies must also equal ( frac{E - B}{2} ), where ( E = 12n^2 ). Therefore:[frac{15n^2}{2} = frac{12n^2 - B}{2}][15n^2 = 12n^2 - B][B = 12n^2 - 15n^2][B = -3n^2]Still negative. Hmm, this is perplexing. It seems like no matter how I approach it, I end up with a negative number of boundary edges, which is impossible. Does this mean that it's impossible to tile the cube in such a way that each rectangle borders exactly five others?But wait, maybe I'm misunderstanding the problem. Perhaps the condition is that each rectangle borders exactly five others along its edges, not necessarily that each edge borders another rectangle. So, a rectangle can have multiple edges adjacent to the same rectangle, but each adjacency counts as one, regardless of how many edges are shared.Wait, but two rectangles can only share one edge, right? So, each adjacency is along one edge. Therefore, if a rectangle borders five others, it must have five edges adjacent to other rectangles, but since it only has four edges, this seems impossible.Unless... unless the rectangle is at a corner where multiple faces meet, and it can share edges with multiple rectangles on different faces. But even then, each edge can only border one other rectangle. So, if a rectangle is at a corner, it can have edges on three different faces, but each edge can only border one rectangle. So, it could border up to three rectangles, one on each adjacent face, plus one on the same face. That would still only be four adjacencies.Wait, maybe the rectangle is placed in such a way that it spans across multiple faces. For example, if a rectangle is placed along an edge where two faces meet, it could potentially border rectangles on both faces. But in that case, the rectangle would still only have four edges, each of which can border at most one rectangle. So, even if it's spanning two faces, it can only border four rectangles at most.This seems to confirm that it's impossible for a rectangle to border five others because it only has four edges. Therefore, the condition that each rectangle borders exactly five others cannot be satisfied for any ( n ).But that contradicts the initial thought that maybe ( n = 2 ) works. Let me double-check that. For ( n = 2 ), the cube has six faces, each with four squares. Covering each face with two ( 1 times 2 ) rectangles, totaling twelve rectangles. Each rectangle on a face borders one other rectangle on the same face. But since the cube is small, each rectangle is at a corner and might also border rectangles on adjacent faces.Wait, in a ( 2 times 2 times 2 ) cube, each rectangle is on a face, but the edges of the rectangles are also edges of the cube. So, a rectangle on the top face shares its edges with the front, right, back, and left faces. But those edges are boundaries of the cube, not adjacent to other rectangles. Therefore, each rectangle only borders one other rectangle on its face, and the other edges are on the boundary. So, each rectangle only borders one other rectangle, not five. Therefore, ( n = 2 ) doesn't work either.This makes me think that maybe no value of ( n ) satisfies the condition because it's impossible for a rectangle to border five others given that it only has four edges. Therefore, the answer might be that it's impossible for all ( n ).But the problem asks for which ( n ) it's possible, implying that there might be some ( n ) where it works. Maybe I'm missing something. Perhaps the tiling isn't just on the faces but also wraps around the edges or something. But no, the problem specifies covering the surface, so it's just the outer faces.Wait, maybe the rectangles can overlap edges and corners in a way that allows them to border more rectangles. For example, if a rectangle is placed along an edge where three faces meet, it could potentially border rectangles on all three faces. But even then, it would still only have four edges, each of which can border one rectangle. So, it could border up to four rectangles, one on each face it's adjacent to, but not five.Alternatively, maybe the rectangles are placed in a way that they share corners with multiple rectangles, but adjacency along a corner doesn't count as bordering along an edge. So, that wouldn't help.I'm starting to think that it's impossible for any ( n ) because of the four-edge limitation. Each rectangle can only border four others, so requiring five is impossible. Therefore, the answer is that it's impossible for all ( n ).But the initial problem seems to suggest that there might be a solution for some ( n ), especially since it's asking for which ( n ). Maybe I'm misunderstanding the problem. Perhaps the condition is that each rectangle borders exactly five others in terms of face adjacency, not edge adjacency. But the problem specifically says "along its edges," so it must be edge adjacency.Wait, maybe the cube is being considered in a way that each rectangle can border multiple rectangles along a single edge. For example, if a rectangle is placed along an edge where multiple rectangles meet, it could border multiple rectangles along that edge. But in reality, each edge can only border one other rectangle because two rectangles can't share the same edge. So, that doesn't help either.I'm really stuck here. It seems like the problem is impossible because of the four-edge limitation. Therefore, my conclusion is that it's impossible to cover the surface of an ( n times n times n ) cube with ( 1 times 2 ) rectangles such that each rectangle borders exactly five others for any ( n ).But wait, maybe I'm missing a clever tiling pattern that allows rectangles to border more than four others by somehow sharing edges in a non-standard way. But I can't think of any such pattern. Each edge can only be shared with one other rectangle, so four edges mean four adjacencies at most.Therefore, I think the answer is that it's impossible for all ( n ). There is no value of ( n ) for which this tiling is possible because it violates the edge adjacency constraint.

💡Alright, so I've got this problem here where I need to find the range of possible values for the expression ( ab + 2bc + 2ca ) given that the real numbers ( a ), ( b ), and ( c ) satisfy the equation ( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 ). The options provided are A: ( (-infty, 4] ), B: ( [-4, 4] ), C: ( [-2, 4] ), and D: ( [-1, 4] ). First, I want to understand the constraint given by the equation ( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 ). It seems like an ellipsoid in three-dimensional space because it's a quadratic equation in three variables. The coefficients in front of ( a^2 ) and ( b^2 ) are both ( frac{1}{4} ), which means that along the ( a ) and ( b ) axes, the ellipsoid extends to ( pm 2 ) since ( frac{1}{4}a^2 = 1 ) implies ( a = pm 2 ). Similarly, along the ( c ) axis, the ellipsoid extends to ( pm 1 ) because ( c^2 = 1 ) gives ( c = pm 1 ).Now, the expression I need to analyze is ( ab + 2bc + 2ca ). This looks like a quadratic form, and I might need to use some inequality techniques or maybe even Lagrange multipliers to find its maximum and minimum values under the given constraint.Let me think about how to approach this. One method that comes to mind is completing the square or using the Cauchy-Schwarz inequality. Alternatively, I could consider using Lagrange multipliers since this is an optimization problem with a constraint.But before jumping into calculus, maybe I can manipulate the given equation to express it in a form that relates to the expression ( ab + 2bc + 2ca ). Let me try that.Starting with the given equation:[frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1]If I multiply both sides by 4 to eliminate the fractions:[a^2 + b^2 + 4c^2 = 4]Hmm, that looks a bit cleaner. Now, I need to relate this to ( ab + 2bc + 2ca ). Maybe I can express ( ab + 2bc + 2ca ) in terms of squares or something that can be bounded using the given equation.Another idea is to consider the expression ( ab + 2bc + 2ca ) and see if I can write it as a combination of squares or use some inequality to bound it.Let me try to group the terms:[ab + 2bc + 2ca = ab + 2c(b + a)]Hmm, that might not be directly helpful. Maybe I can factor it differently or complete the square.Alternatively, perhaps I can use the Cauchy-Schwarz inequality. The Cauchy-Schwarz inequality states that for any real vectors ( mathbf{u} ) and ( mathbf{v} ), we have:[|mathbf{u} cdot mathbf{v}| leq |mathbf{u}| |mathbf{v}|]But I'm not sure how to apply this directly here because ( ab + 2bc + 2ca ) isn't a straightforward dot product. Maybe I can represent it as a dot product somehow.Wait, another approach is to use the method of Lagrange multipliers. Let me set up the function to maximize and minimize:Let ( f(a, b, c) = ab + 2bc + 2ca )Subject to the constraint ( g(a, b, c) = frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 - 1 = 0 )The method of Lagrange multipliers tells us that at the extrema, the gradient of ( f ) is proportional to the gradient of ( g ). So, we have:[nabla f = lambda nabla g]Calculating the gradients:[nabla f = left( frac{partial f}{partial a}, frac{partial f}{partial b}, frac{partial f}{partial c} right) = (b + 2c, a + 2c, 2b + 2a)][nabla g = left( frac{partial g}{partial a}, frac{partial g}{partial b}, frac{partial g}{partial c} right) = left( frac{1}{2}a, frac{1}{2}b, 2c right)]So, setting up the equations:1. ( b + 2c = lambda cdot frac{1}{2}a )2. ( a + 2c = lambda cdot frac{1}{2}b )3. ( 2b + 2a = lambda cdot 2c )And the constraint:4. ( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 )Now, I need to solve this system of equations. Let me write them out again:1. ( b + 2c = frac{lambda}{2} a ) --> Multiply both sides by 2: ( 2b + 4c = lambda a )2. ( a + 2c = frac{lambda}{2} b ) --> Multiply both sides by 2: ( 2a + 4c = lambda b )3. ( 2b + 2a = 2lambda c ) --> Divide both sides by 2: ( b + a = lambda c )4. ( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 )So, from equation 3: ( a + b = lambda c )Let me denote this as equation 3.From equation 1: ( 2b + 4c = lambda a )From equation 2: ( 2a + 4c = lambda b )Let me try to express ( lambda ) from equations 1 and 2.From equation 1: ( lambda = frac{2b + 4c}{a} )From equation 2: ( lambda = frac{2a + 4c}{b} )So, setting them equal:[frac{2b + 4c}{a} = frac{2a + 4c}{b}]Cross-multiplying:[(2b + 4c) b = (2a + 4c) a]Expanding both sides:Left side: ( 2b^2 + 4bc )Right side: ( 2a^2 + 4ac )So, setting them equal:[2b^2 + 4bc = 2a^2 + 4ac]Divide both sides by 2:[b^2 + 2bc = a^2 + 2ac]Bring all terms to one side:[b^2 - a^2 + 2bc - 2ac = 0]Factor:[(b - a)(b + a) + 2c(b - a) = 0]Factor out ( (b - a) ):[(b - a)(b + a + 2c) = 0]So, either ( b - a = 0 ) or ( b + a + 2c = 0 ).Case 1: ( b = a )Case 2: ( b + a + 2c = 0 )Let's analyze both cases.**Case 1: ( b = a )**If ( b = a ), then from equation 3: ( a + a = lambda c ) --> ( 2a = lambda c )From equation 1: ( 2b + 4c = lambda a ). Since ( b = a ), this becomes:( 2a + 4c = lambda a )From equation 3, ( lambda = frac{2a}{c} ). Substitute into equation 1:( 2a + 4c = frac{2a}{c} cdot a )Simplify:( 2a + 4c = frac{2a^2}{c} )Multiply both sides by ( c ):( 2a c + 4c^2 = 2a^2 )Divide both sides by 2:( a c + 2c^2 = a^2 )Rearrange:( a^2 - a c - 2c^2 = 0 )This is a quadratic in ( a ):( a^2 - a c - 2c^2 = 0 )Using quadratic formula:( a = frac{c pm sqrt{c^2 + 8c^2}}{2} = frac{c pm 3c}{2} )So, ( a = frac{c + 3c}{2} = 2c ) or ( a = frac{c - 3c}{2} = -c )So, two subcases:Subcase 1a: ( a = 2c )Subcase 1b: ( a = -c )Let's analyze Subcase 1a: ( a = 2c )Since ( b = a = 2c ), we can substitute into the constraint equation:( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 )Substitute ( a = b = 2c ):( frac{1}{4}(4c^2) + frac{1}{4}(4c^2) + c^2 = 1 )Simplify:( c^2 + c^2 + c^2 = 1 ) --> ( 3c^2 = 1 ) --> ( c^2 = frac{1}{3} ) --> ( c = pm frac{sqrt{3}}{3} )Thus, ( a = b = 2c = pm frac{2sqrt{3}}{3} )Now, compute ( ab + 2bc + 2ca ):Since ( a = b = 2c ):( ab = (2c)(2c) = 4c^2 )( 2bc = 2(2c)c = 4c^2 )( 2ca = 2c(2c) = 4c^2 )So, total:( ab + 2bc + 2ca = 4c^2 + 4c^2 + 4c^2 = 12c^2 )But ( c^2 = frac{1}{3} ), so:( 12 times frac{1}{3} = 4 )Therefore, in Subcase 1a, the expression equals 4.Subcase 1b: ( a = -c )Since ( b = a = -c ), substitute into the constraint:( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 )Substitute ( a = b = -c ):( frac{1}{4}c^2 + frac{1}{4}c^2 + c^2 = 1 )Simplify:( frac{1}{2}c^2 + c^2 = 1 ) --> ( frac{3}{2}c^2 = 1 ) --> ( c^2 = frac{2}{3} ) --> ( c = pm frac{sqrt{6}}{3} )Thus, ( a = b = -c = mp frac{sqrt{6}}{3} )Compute ( ab + 2bc + 2ca ):( ab = (-c)(-c) = c^2 )( 2bc = 2(-c)c = -2c^2 )( 2ca = 2c(-c) = -2c^2 )So, total:( ab + 2bc + 2ca = c^2 - 2c^2 - 2c^2 = -3c^2 )Since ( c^2 = frac{2}{3} ):( -3 times frac{2}{3} = -2 )Therefore, in Subcase 1b, the expression equals -2.**Case 2: ( b + a + 2c = 0 )**From equation 3: ( a + b = lambda c )But from Case 2, ( a + b = -2c ). So, ( -2c = lambda c )Assuming ( c neq 0 ), we can divide both sides by ( c ):( -2 = lambda )So, ( lambda = -2 )Now, from equation 1: ( 2b + 4c = lambda a = -2a )So, ( 2b + 4c = -2a ) --> ( 2b + 4c + 2a = 0 ) --> ( a + b + 2c = 0 ), which is consistent with Case 2.Similarly, from equation 2: ( 2a + 4c = lambda b = -2b )So, ( 2a + 4c = -2b ) --> ( 2a + 4c + 2b = 0 ) --> ( a + b + 2c = 0 ), again consistent.So, in this case, we have ( a + b + 2c = 0 ) and ( lambda = -2 )Now, let's express ( a ) in terms of ( b ) and ( c ):From ( a + b + 2c = 0 ), ( a = -b - 2c )Substitute this into the constraint equation:( frac{1}{4}a^2 + frac{1}{4}b^2 + c^2 = 1 )Substitute ( a = -b - 2c ):( frac{1}{4}(-b - 2c)^2 + frac{1}{4}b^2 + c^2 = 1 )Expand ( (-b - 2c)^2 ):( b^2 + 4bc + 4c^2 )So, substitute back:( frac{1}{4}(b^2 + 4bc + 4c^2) + frac{1}{4}b^2 + c^2 = 1 )Simplify each term:( frac{1}{4}b^2 + bc + c^2 + frac{1}{4}b^2 + c^2 = 1 )Combine like terms:( left( frac{1}{4}b^2 + frac{1}{4}b^2 right) + bc + left( c^2 + c^2 right) = 1 )Simplify:( frac{1}{2}b^2 + bc + 2c^2 = 1 )Now, this is a quadratic in ( b ) and ( c ). Let's see if we can find the extrema.But perhaps it's easier to express ( b ) in terms of ( c ) or vice versa.Alternatively, let's express the expression ( ab + 2bc + 2ca ) in terms of ( b ) and ( c ).Given ( a = -b - 2c ), substitute into ( ab + 2bc + 2ca ):First, compute each term:( ab = (-b - 2c)b = -b^2 - 2bc )( 2bc = 2bc )( 2ca = 2c(-b - 2c) = -2bc - 4c^2 )Now, add them up:( ab + 2bc + 2ca = (-b^2 - 2bc) + 2bc + (-2bc - 4c^2) )Simplify:- ( -b^2 - 2bc + 2bc - 2bc - 4c^2 )Combine like terms:- ( -b^2 - 2bc - 4c^2 )So, ( ab + 2bc + 2ca = -b^2 - 2bc - 4c^2 )Now, we have the constraint equation from Case 2:( frac{1}{2}b^2 + bc + 2c^2 = 1 )Let me denote this as equation (A).We need to express ( ab + 2bc + 2ca = -b^2 - 2bc - 4c^2 ) in terms of equation (A).Let me see:From equation (A):( frac{1}{2}b^2 + bc + 2c^2 = 1 )Multiply both sides by 2:( b^2 + 2bc + 4c^2 = 2 )Notice that ( ab + 2bc + 2ca = -b^2 - 2bc - 4c^2 = -(b^2 + 2bc + 4c^2) = -2 )Wait, that's interesting. So, ( ab + 2bc + 2ca = -2 ) in this case.But wait, let me double-check:From equation (A):( frac{1}{2}b^2 + bc + 2c^2 = 1 )Multiply by 2:( b^2 + 2bc + 4c^2 = 2 )So, ( b^2 + 2bc + 4c^2 = 2 )Thus, ( ab + 2bc + 2ca = - (b^2 + 2bc + 4c^2) = -2 )So, in Case 2, regardless of the values of ( b ) and ( c ), as long as they satisfy the constraint, the expression ( ab + 2bc + 2ca ) equals -2.Wait, that seems a bit strange. Let me verify.Given ( a = -b - 2c ), then:( ab + 2bc + 2ca = (-b - 2c)b + 2bc + 2c(-b - 2c) )Compute each term:1. ( (-b - 2c)b = -b^2 - 2bc )2. ( 2bc = 2bc )3. ( 2c(-b - 2c) = -2bc - 4c^2 )Adding them together:( -b^2 - 2bc + 2bc - 2bc - 4c^2 = -b^2 - 2bc - 4c^2 )Which is indeed equal to ( - (b^2 + 2bc + 4c^2) ). And since ( b^2 + 2bc + 4c^2 = 2 ), this expression is ( -2 ).So, in Case 2, the expression ( ab + 2bc + 2ca ) is always -2.Wait a minute, that's interesting. So, in Case 2, regardless of the specific values of ( b ) and ( c ), as long as they satisfy the constraint, the expression is fixed at -2.But in Case 1, we found that the expression can be either 4 or -2.So, putting it all together, the maximum value of the expression is 4, and the minimum value is -2.Therefore, the range of possible values for ( ab + 2bc + 2ca ) is from -2 to 4, inclusive.So, the correct answer should be option C: ( [-2, 4] ).Just to make sure I didn't make any mistakes, let me recap:- I used Lagrange multipliers to set up the system of equations.- Solved for the cases where either ( b = a ) or ( b + a + 2c = 0 ).- Found that in the case ( b = a ), the expression can be 4 or -2.- In the case ( b + a + 2c = 0 ), the expression is always -2.- Therefore, the maximum is 4 and the minimum is -2.Yes, that seems consistent. So, I'm confident that the range is ( [-2, 4] ), which is option C.