Webinars SET B - Grade 9-10 - Sunday@6-7pm EST-Recording Webinar 5

Video Transcription

Video Summary

The lesson focuses on finding minimum and maximum values through problem-solving with sequences and patterns, building on previous lessons. It begins with a warm-up involving three-digit consecutive numbers where each term has at least one odd digit, exploring the longest such sequence. By trial and error and logical reasoning, the longest sequence is found to be from 289 to 399, totaling 111 terms.<br /><br />The instructor then distinguishes straightforward minimization/maximization problems, like quadratic functions, from more complex optimization challenges requiring construction and proof, such as map-coloring problems or arranging colored cells in grids.<br /><br />Several practice problems involve number theory and divisibility, including constructing the longest list of numbers 1 to 10 where each adjacent pair has a divisor relationship. A solution involving 9 numbers is found by carefully grouping numbers with shared factors.<br /><br />Next, a fraction problem uses numbers 1 to 22 to form pairs whose fractions yield integer values, emphasizing strategic pairing of primes and multiples.<br /><br />The lesson progresses to geometric and combinatorial optimization, such as coloring rectangular grids with black and white squares to maximize sum values relating to adjacency, analyzing factorizations of large numbers (2018) to choose grid dimensions for optimal coloring patterns.<br /><br />Another interesting challenge involves arranging numbers 1 to 10 around a circle so the smallest sum of each number and its neighbors is maximized. Through averaging and trial, the maximum smallest sum is near 15, achieved by careful number placement to balance sums.<br /><br />Finally, a large-scale combinatorial problem considers planting 100 trees (oaks and birches) so that no two oaks have exactly five trees between them. By analyzing spacing patterns and constraints in groups of 12, it is concluded that at most 52 oaks can be planted, supported by logical grouping and divisor reasoning.<br /><br />Overall, the lesson illustrates creative problem-solving strategies: building examples, using divisibility, strategic trial and error, constructing solutions that meet constraints, and logical arguments to establish minima or maxima in complex settings.

Keywords

minimum and maximum values

sequences and patterns

odd digits in numbers

optimization problems

number theory and divisibility

fraction pairing

geometric and combinatorial optimization

coloring rectangular grids

arranging numbers in a circle

planting trees with spacing constraints