Batch 2 - Class 72 - Coloring Cubes & Spatial Visualization II

• A 5 by 5 by 5 cube is formed using identical 1 by 1 by 1 cubes. A number of the smaller cubes are removed by punching out the fifteen designated columns from front to back, top to bottom, and side to side. What percentage of the volume of the original cube remains following the removal of the fifteen columns?

• Answer 54.4% (Only 68 of surface cubes remain - inside is hollowed)

• Suppose all six faces of an n × n × n are painted red. Then one of the n 3 unit cubes is randomly selected and tossed like a die. What is the probability that the face obtained is painted? 

  Answer: 1/n. There are 6n^2 painted faces out of total of 6n^3 (the problem can also be solved by taking all cases of cubes with different number of painted sides)
• Bob and Ann play the following game with 8 white unit cubes. Ann wins if she can assemble a 2 × 2 × 2 cube that has only white faces exposed. But Bob gets to paint four of the 8 · 6 = 48 white faces black. Who wins? 

  Answer: Bob - if he paints 4 sides of a single cube, then one can't assemble them into a white 2x2x2 cubeHow about a 3x3x3 cube - how many faces need to be painted?Answer 12 - the middle one, and then one more with all sides
•  Three cubes with volumes 1, 8, and 27 are glued together to obtain a solid polyhedron with minimal surface area. What is that surface area?

  Answer: The combined surfaces of the three cubes is 6(9 + 4 + 1) = 6·14 = 84. We can reduce the surface area of the polyhedron by 2·4 = 8 when we put the two larger cubes together. Then we can reduce the surface area of the polygon by 4 more by gluing the unit cube so that it has a face in common with each of the other two cubes. So the minimal surface area of the polyhedron is 84 − 8 − 4 = 72 square units. 

• Ask kids if they know what is gravity assist and how it works

  Explain (Use http://solarsystem.nasa.gov/basics/grav/primer.php) including the planet example , downhill/uphill and  train exampleExample: New Horizons flight to pluto used gravity assist from Jupiter, accelerating it by 4000 m/s - without gravity assist, it would have taken 2-4 years more to reach Pluto (and more fuel)Launched Jan 2006, Closest to Jupiter Feb 2007, July 2015 started exploring Pluto, will explore Kuiper belt object Jan 2019 (Kuiper belt is region of solar system beyond neptune, and including pluto, like an asteroid ring)
• So what would be the effect of a gravity assist fly by on Jupiter

  It would slow down the planet slightly, just like the train would slow down slightly in previous exampleJupiter is 1.9x10^27 kgs; New Horizons 500 kgSo 4x10^24 ratioIf New Horizons gained 4000 m/s, then Jupiter would have lost about 10^(-21)m/s (the actual math is more complex, but this works for our purpose)This does change Jupiter's speed, and makes the year shorter there - but only by few dozen nanoseconds by the time the sun flames out in 5 billion yearsSo, how many flybys would it take to make any significant difference?New Horizons mission cost US govt $700M over 2001-2016Over this time US govt spend $48000 Billion for other things (well, all other things)So if we diverted all that spend into launching New Horizons, we could launch about 70,000 of thoseThat has other repercussions, in terms of amount of nuclear fuel requiredWorse, the chances of accidental radioactive release on these launches is about 1/300, so there would be 200 of those to deal with!With all this put together, we could shave off 2 ms off Jupiters calendar over next 5 Billion years!Actually, earth only weighs (6x10^24 kg) a fraction of Jupiter (1/300) so even if we flung the whole earth at Jupiter, it would only slow down by a fraction of percentage - few dozen miles an hourJust like if you wanted to stop the train by throwing tennis balls at it!

• (Geoffrey - 87) A kid got 100% marks in geometry. Her elder sister remarked, "I suppose you think you know geometry from A to Z." "Sure I do." "Prove it, then." The elder sis drew a big A and a big Z, and asked the younger one to divide the A into four parts, which can be put together to make the Z, and without turning over any of the pieces. Can yo do that?

• Answer: