Key words: Geometry, Unusual geometry, Math, Physics, Chemistry, High school, Geometry Olympiads, Free Geometry, Euclidean Geometry, Calculus, Geometric Construction. Oh yes, going-nowhere discussions, haikus, and poems too.

## Saturday, March 26, 2011

### Fox 336

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## Thursday, March 24, 2011

### Fox 335

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## Monday, March 21, 2011

### Fox 334

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## Monday, March 14, 2011

### Fox 333

**Red Fox**:

**Happy Pi+1 Day!**

**Himalayan Fox**: What is that exactly?

**Red**:

**Well, March 14 was the Pi Day, you know, 3.14, and today is March 15th.**

**Himalayan:**But Pi+1 would be 4.14 which is April 14th, isn't it?

**Red**:

**Oh, I didn't think that way. Whatever it is, happy March 15th to you brother.**

**Himalayan:**OK, I'll mechanically say "to you too", but can we really happy while thousands of souls swept away with the water?

**Red**:

**Well, we can not die with the dead.**

**Himalayan:**But we can help the living.

**Red**: I hear you brother.

**Himalayan:**Then we can celebrate 2Pi Day a few months later.

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## Wednesday, March 9, 2011

### Balanced Mancala Problem

We rarely ask pure math problems but here is one with a "very little" touch of geometry. This does not need to be defined as a mancala game, but here it goes:

We have stones coming in batches. Each stone has a color and a weight. If the color of a stone is:

** Yellow: **It must be placed every other pit (every 2 pits) Batch size is always: 6 Total batches: y Total number = 6y

** Red: **Must be placed every 3 pits Batch size: 4 Total batches: r Total number = 4r

** Green: **Must be placed every 4 pits Batch size: 3 Total batches: g Total numbers = 3g

** Blue: **Must be placed every 6 pits Batch size: 2 Total batches: b Total number = 2b

** Purple: **Must be placed only once in 12 pits Batch size: 1 Total batches: p Total numbers = p

So there are **N=6y+4r+3g+2b+p** many stones. The stones in the same batch have the same weight. Different batches may have different weights. WLOG, assume that all weights are integers. We have a proof that ending up with the best well-balanced mancala is very difficult (NP-Hard). Here "well-balanced" means that the pit with the maximum weight is minimized when all stones are distributed. Let's call this maximum pit weight as **W**.

__Consider the following heuristic process: __

**Step 1**. Sort the batches with respect to their weights (batches with the high-weight stones go first) **Step 2**. Insert the first batch starting from pit number 1. **Step 3**. Insert the next batch in a way that the total maximum weight throughout 12 pits remains minimum. **Step 4**. Repeat Step 3 until all batches are placed in the mancala. Let **H **be the maximum weight throughout 12 pits.

** A simple Example:** Suppose we have only 4 batches: Yellow (6 stones, each 45 grams) Blue (2 stones, each 40 grams) Yellow (6 stones, each 30 grams) Green (3 stones, each 20 grams) First batch (Yellow) goes to pits: 1, 3, 5, 7, 9, and 11. H=45. Second batch (Blue) goes to pits: 2 and 8. H=45. Third batch (Yellow) goes to pits: 2, 4, 6, 8, 10, and 12. H=70. Fourth batch (Green) goes to pits: 3, 7, and 11. H=70.

In this exercise, heuristic actually finds the optimum, i.e., H=W=70 grams, observed in pits 2 and 8.

** And the question:** Prove that the worst-case of the heuristic solution,

**H**, is always less than

**2W**. In other words,

**W ≤ H ≤ 2W**always holds. If you disagree, then try to generate a counter-example.

Good luck!

## Monday, March 7, 2011

### Fox 332

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