Batch 3 - Class 116 - Stacking Pillows
Pre-Class Problem:
A boat is carrying certain number of fish tanks (up to maximum of 13 tanks), each with equal number of fish. The boat hits a storm that knocks off all the tanks, and they land in three sectors A and B. Imaging shows that sector A has 4 tanks and 2 sharks, whereas sector B has 2 tanks and 4 sharks. The Imaging machine is unable to sense sector C. However, it is known that total fish in those areas (including those in tanks and the sharks) are 50, each sector has at least 1 shark and upto 7, and that no two sectors can have the same number of sharks. How many tanks have landed in sector C?
- Answer: 7 (Total fish in tanks will be 39 in 13 tanks)
Attendance: Muskaan, Tishyaa, Nandini, Aneyaa, Khushi, Arnav, Anishka, Anshi, Liza, Ahana, Palak, Manya, Arnav, Zorawar, Damini, Siddhant, Diya Bansal
Class Notes:
A king was reclining on his fifteen pillows, trying to get comfortable - and failing. He suddenly realized the reason - the stack of pillows needed to decrease in height from left to right.
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Can you help him fix that? One possible solution is as under
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Suddenly, a challenger came by, and told the king, "I can make a superior pile of pillows"
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Is this a valid pile of pillows? No, the stacks all need to be of different heights. So the challenger rearranged them.
Is this a valid pile of pillows? No, the stacks need to decrease in size left to right. Can you help the challenger create a "better" pile of pillows?
- What is better - kids should ask this question. We will define better as follows - we will go left to right comparing the size of stacks, and whoever wins more stacks has a better pile
- If kids raise the question of "tie" here, clarify that in case of tie, the king wins that stack
Let kids come up with better piles. Let them know that they dont need to have five stacks - they can have fewer. Pick the following for example:
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Now, this one beats the king. But the king decides to steal the challenger's pile. Can you now help the challenger come up with an even better one?
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One of the possible solutions above! But the king steals that one too! What should the challenger do now? Can you help him come up with an even better one?
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Oops! What has happened above? The original pile is better than this last one...
- What is the notion of better?
- Are there other examples kids can come up with where A>B, B>C doesn't really imply A>C?
- This relationship is called transitivity. ">" is a transitive relationship
- Rock Paper Scissor is a good example of another non-transitive relationship!
Can the king find an arrangement that can not be beaten?
- Here are a few
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Liza's Question: Given 15 pillows, how many different ways exist to stack them up?
- Discussion: We can think about this problem recursively, i.e. think of breaking the larger problem into smaller problems. With 15 pillows, the tallest stack can be anywhere from 1 pillow tall to 15 pillows tall. Given any choice of the first pillow, say 8 pillows, we then know that now we are required to arrange the remaining (15-8)=7 pillows into stacks so that the tallest stack is between 1 to 7 pillows tall. So we have been able to break up F(Pillows, Tallest Stack) into Sum of F(Pillows - Tallest Pillow, Any number from 1 to Tallest Pillow -1). And finally, the number of stacks with 15 Pillows is just Sum of F(15, Any number from 1 to 15). The same has been modeled in attached Excel sheet. In formal math, this may be written as Sum for "tallest" going from 1 to "Pillows" { Sum for "next tallest" going from 0 to "tallest"-1 { Answer for Function ("Pillows"-"tallest", "next tallest") }}. By expressing the answer in terms of smaller problems, we can now start to solve the smaller problems and compose the solution
- The answer for 15 pillows seems to be 27 (to be verified). Note that the corresponding solutions can be found by following a trail on the Excel sheet attached - for example, the one highlighted in yellow can be read off as (8,4,2,1) by following the formula dependencies, and looking at corresponding row numbers.
Homework
- Petals around the roses - explain the puzzle to kids by rolling 5 dice, and answering them the code number in the roll of 5 dice. Take the first ten rolls, and ask them to note the rolls down along with answers. Ask them to come back next week with a solution to predict the answer for any given roll. You may pick the rolls of dice at http://web.archive.org/web/20070509082215/http://www.borrett.id.au/computing/petals-bg.htm
References: