Batch 3 - Class 163 - 15 Card Difference Triangle
Pre-Class Exercise
- I have a 24L of water in a jar. Can you divide into 3 equal parts using only a 13L, a 5L and an 11 liter Jar and of course 24L jar.
- Answer
24 13 11 5
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24 0 0 0
11 13 0 0
6 13 0 5
6 2 11 5
8 0 11 5
8 5 11 0
8 13 3 0
8 8 3 5
8 8 8 0
Attendance Arnav, Anishka, Damini, Siddhanth, Arjun
Class puzzles
I have 15 cards, numbered consecutively from 1 to 15. I want to lay them out in a triangle. I’ve put numbers on top three for the later reference:
However, I do not want any old arrangement. I want each card to be the difference between the two cards immediately below it, to left and right. For example, 5 is the difference between 4 and 9. The differences are always calculated so that they are positive. This condition does not apply to the cards in the bottom row, you appreciate.
The top three cards are already in place – and correct. Can you find how to place the remaining 12 cards?
- Younger kids may start with 3 cards, then 6 cards and then 10 cards
- This is essentially a hit and trial problem. So kids should look to discover heuristics that make the hit and trial efficient. Some of these are as under:
- The largest number (15 in this case) must lie on bottom row, since that number can't be formed with a difference. So the largest number in second bottom row can be 14. Similarly, 13 can't be in third bottom row (because then we would need two 1's), so the largest number in middle row can be 12.
- The smallest numbers also must lie in lower rows because we will need some of them for differences amongst largest numbers
- There is a number in lower row which is sum of a number above and a smaller number next to it. So there must be a large number in bottom row, which is sum of 5 numbers, one in each row. That number can only be 15, and those 5 numbers 1,2,3,4,5 must be in different rows.
- Now if we take the above, we can start to build chains using the largest numbers in any given row. For example, 5, then 9 (and we have consumed 5 and 4 amongst smaller numbers). Now we could have a 10, 11 or 12 below 9 by consuming another small number and so on. About 12 combinations to check for in all.
- Also note that there can't be a 3-2-1 subtriangle, which does not have any of the 1,2,3 numbers. This is because the smallest differences in these can be 6,7,8 on bottom row of that sub triangle. Thus, 15 must lie at center of bottom row, and numbers 1,2,3 can't all lie at one "side" of the big triangle.
Answer:
Note that it has been proven that such a triangle does not exist for 21 numbers.
Homework Problem:
- How does a bishop behave on a torus chess board?
References: