Three fifths or four sevenths?
Continue the pattern... could this help you decide?
Is this approach always possible?
Three fifths or four sevenths? Continue the pattern... could this help you decide? Is this approach always possible?
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In Being Peace, Thich Nhat Hanh asks: Can we be a river and experience the hopes and fears of a river? Before I arrived, the river was undifferentiated energy, like myself. I come near to the river and dwell. In Conversation on a Country Path, Heidegger describes thinking as a ‘lettingoneselfintonearness’. I watch the water flow along its path, I listen to the gentle bubbling sound. In this act of attention, there is differentiation: between the river and its surroundings, and between the river and me. But there is also an attraction, a sense of wholeness. In The Principles of Psychology, William James states: …each of us literally chooses, by his ways of attending to things, what sort of a universe he shall appear to himself to inhabit. A river is never the same, but we consider it the same if it follows a recognisable path. With recognition comes identity. We may gain our identities in a similar way, as described by James: A man has as many social selves as there are individuals who recognise him. Can we be a river and experience the hopes and fears of a river? Does the river have hopes and fears, a sense of self? From where do we derive our sense of self? William James says ‘the stream of our thought is like a river’, and describes consciousness as: … a succession of perishing thoughts, endowed with the functions of appropriation and rejection … Thoughts flow past, but sometimes we bring our attention to the flow itself. This may be regarded as stepping out of the flow of time and becoming aware of time itself. We cannot directly experience the past or future, only the present. But even the present is elusive, as James describes: Let any one try, I will not say to arrest, but to notice or attend to, the present moment of time. One of the most baffling experiences occurs. Where is it, this present? It has melted in our grasp, fled ere we could touch it, gone in the instant of becoming. He suggests that the present moment is an abstraction, and that we experience the present not as a moment, but as an extended period of time which he calls the ‘specious present’: These lingerings of old objects, these incomings of new, are the germs of memory and expectation, the retrospective and the prospective sense of time. The brainprocesses that create this feeling of succession are those that give our sense of time, and ultimately our sense of a continuing self: Resemblance among like parts of a continuum of feelings (especially bodily feelings)… constitutes the real and verifiable personal identity which we feel… Assuming that the pattern continues 'as shown', and that there must be at least one red in a line, find the minimum and maximum possible fractions of white and red. Same question for this pattern: Explore similar patterns of your own. Can you predict the minimum and maximum fractions for any pattern? In my excitement about this upcoming ATM book, I've been playing with Cuisenaire rods.
Each diagram below shows three terms taken from a sequence (click on the image to see more). The question is: How would *you* extend the sequence (in both directions) in each case? [Do any problems arise with your way of extending each sequence?] May I invite you to watch this video, which shows that 13 can be written as 2^2 + 3^2, which is the sum of two consecutive Fibonacci numbers squared ('Fibonacci squares'). Q1. Which other terms of the Fibonacci sequence can be written as the sum of two consecutive Fibonacci squares? [why?] Q2. Show that the rest of the Fibonacci sequence can be written as the difference of two Fibonacci squares... [and deduce that these terms are also multiples of some Fibonacci number (apart from 1)]. Q3. Show that difference of two consecutive Fibonacci squares can be written as the product of the two Fibonacci numbers on either side (of the consecutive ones). For example, 8^2  5^2 = 3 x 13. Q4. Here's an interesting pattern I discovered after playing around with Cuisenaire rods: 8^2 + 2^2 = 2.(3^2 + 5^2) 13^2 + 3^2 = 2.(5^2 + 8^2) ... Can you find any other interesting patterns like this? On getting excited about this new ATM book about Cuisenaire rods, I made the following puzzle involving nontraditional rods: Can you deduce the lengths of the five rods? [The grey dotted lines represent coincident edges] Choose any nonprime number N, and then choose two of its factor pairs. For example, if we choose 20, we may take the factor pairs (4,5) and (2,10). Now add all of these factors together (for this example this would be 4 + 5 + 2 + 10 = 21). Find an example for which this sum is prime... or explain why it is not possible.  If you would like a hint, here is a video I made which may give some insight. 
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