Batch 2 - Class 70 - Building Blocks
Preclass Exercise:
- (Matt Enlow's Rectangle Division Puzzle) What is the fewest number of squares into which you can divide an 11 x 12 rectangle? What about a 12 x 13 rectangle? What about an 11 x 13 rectangle?
- Answer:
Attendance: Anisha, Tishyaa, Aryan, Diya, Adit, Abhiram, Liza, Nandini, Praharsh, Khushi
Class Notes:
Building Blocks
You have an unlimited supply of cubes. Each cube’s side length is a number of the form 2^i , where i is any nonnegative integer. That is, there are cubes with edges of length 1 unit, 2 units, 4 units, 8 units, 16 units, and so on. You wish to build a tower of a given exact height by stacking cubes one on top of another.
Part 1: Building Three Towers - You need to build three towers, of heights 10, 15, and 32.
- What is the biggest number of cubes you could use to build the three towers?Answer: All 1s, so 10+15+32=57
- What is the smallest number of cubes you could use to build the three towers?Answer: Binary, so 2+4+1=7
- What is the biggest possible total volume of the three towers?Answer: Biggest volume is using biggest cubes, so (8^3+2^3)+(8^3+4^3+2^3+1^3)+(32^3)=33873
- What is the smallest possible total volume?Answer: All 1s, so 57
- What is the largest possible total surface area for the three towers?Answer: Its easy to calculate total surface area (7062). For hidden surface area, let kids try different arrangements10 units - sequence is 2-8 so hidden area of 1215 units - sequence is 2-4-1-8 so hidden area is 1632 units - hidden area is 1024So total is 7062-12-16-1024=6010 sq units
- What is the smallest possible total surface area?Answer: Using all 1s. Total surface area is 57*6(1^2)=342. Calculate hidden (19, 29, 63 resp), so net 231 sq units
Part 2: Limited Supplies for One Tower Now you have only m cubes, of edge length N1, N2, N3, …, Nm. For simplicity we’ll suppose they go in increasing order, so N1 is the smallest and Nm is the biggest. There may be ties (or they may even all be the same size). You use them all to build a single tower.
- What is the height of this tower?Answer: N1+N2+...+Nm
- What is the total volume of this tower?Answer: N1^3+N2^3+...+Nm^3
- What is the maximum possible total surface area for this tower?Answer: Arrangement - Let kids think about what creates max area - its ordering and then moving N1 as second square from top, N2 as 4th and so onComputation - Need to compute total surface area, and then subtract the bottom and overlapsFor example, for 1,2,4,8 sequence is 2,4,1,8 and answer is 6(1^2+2^2+3^2+4^2) - 2^2+2(2^2)+2(1^2)+2(1^2)= 510-16 = 494
- What is the minimum possible total surface area?Answer: Arrangement - stack largest at bottom and then decreasing orderComputation - total is same as before, hidden is moreFor example, for 1,2,4,8 sequence, hidden is 8^2+2(4^2)+2(2^2)+2(1^2)=106 and hence net surface area is 510-106=404
Part 3: Friendly Fun Your friend builds a tower using at most n blocks of each side length. For instance, if n is 3 then your friend selected blocks from a pile with edges of length 1, 1, 1, 2, 2, 2, 4, 4, 4, 8, 8, 8, and so on. Also, the largest block in your friend’s tower has length c, which equals 2^k for some value of k.
- What is the largest possible number of cubes that could be in your friend’s tower?Answer: n(k+1)
- What is the smallest possible number of cubes in your friend’s tower?Answer: 1
- What is the largest possible height for your friend’s tower?Answer: n(2c-1) - May need to explain the sum (1+2+2^2+...+2^k)=(2.2^k-1)
- What is the smallest possible height?1
What If? - What if the Earth had twice the diameter it currently has?
- Effects on gravityHow much would gravity be different. Introduce kids to the formula for gravity2x Gravity - how would this affect plants? Trees?Collapse?Evolve for stronger stems?How would this affect animalsThicker legs?How would this affect humans?Evolutionary process?How would it affect the atmosphere?What else could it affect?Hold! we just assumed the same density! What if the density of earth was half as much?Same gravityMore space to live!What would happen to the moon?Fall closer? Would it fall all the way in?It could be twice as far and have same 24 hour revolutionOr it could be at the same place and have an 8 hour cycle (r^3 is proportional to T^2*Mass)How would this affect tidal waves? Three times a day high tide!Kids could imagine the above with what happens when a marble is revolved in a gravitational spiral
- Now, other things could change besides gravity, and so on... (Imagine that the earth is not even uniform!)
- The Gravitational Waves Discovery! What does it mean?
Homework
- Ten slips of paper, numbered 1 through 10, are placed in a hat. Three numbers are drawn out, one after another. What is the probability that the three numbers are drawn in increasing order?Answer: 1/6
References: