# Bogglers

By Scott Kim|Sunday, September 1, 2002

The Divided Alphabet puzzle here is slightly different from the version published in the magazine. The letter "M" has been redrawn to yield only one correct answer.

Breaking Up Is Hard to Do
Is your brain limber enough to do the splits?

Divided Alphabet
Each letter in the alphabet shown above can be cut into two pieces that are exactly the same shape and size. Cutting only along grid lines and 45-degree diagonals, can you split each letter into its two identical parts? You can rotate and flip the pieces to make them match up. We've given you the answer for the letter D.

Divided Words
Fast and slow, ball and socket, action and reaction, ordinate and abscissa— scientific concepts often come in twos. Unscramble the words and phrases below to find 10 pairs of terms taken from 10 different areas of science. The terms can be antonyms, synonyms, or even well-known counterparts. For example, problem 1, ABC IDEAS, unscrambles to form ACID / BASE, familiar terms from chemistry. To help you untangle these twisted terms, the 10 areas of science appear below but not in the same order as the terms.
 1. ABC IDEAS 2. FOLDOUTS 3. AIR TOWEL 4. FLUTE EARS 5. USUAL DRAIN 6. BLEAK WITCH 7. DOOR SEARCH 8. NIFTY IONIZER 9. PAINT ALL MAN 10. CAVALIER WEPT Acoustics Anatomy Biology Chemistry Cooking Dynamic Systems Logic Mathematics Photography Physics of Light

1. [Easy]     The land of Airdish comprises four towns: Ay, Bee, Cee, and Dee. Six roads link every town to every other town. The Redford Construction Company (red lines in the diagram at right) owns three of the roads, and Blue Devil Highways owns the others. A new law declares that no single company can own three roads that make a triangle connecting three towns. The Redford Construction Company is in violation of the law. Can you change the color of two roads so neither company owns a triangle? Note: Only triangles with cities at all three corners are outlawed; triangles with corners that end at overpasses are legal.

2. [Difficult]     Over time, a city called Eee was built, with four new roads connecting it to the other cities. Can you color the roads red and blue so neither company owns a triangle?

3. [Very Difficult]     Eventually, the city of Eff arose, with five new roads connecting it to the five other cities. Unfortunately, no one was able to find a way to color the roads red and blue without creating a red or a blue triangle. Can you reason out why such a coloring is impossible? Hint: There are five roads from every city, and at least three of these must belong to one company. How does this affect the three towns these roads lead to?

4. [Difficult]     What is the greatest number of roads you can color red or blue on the six-city map without breaking the triangle law?

5. [Difficult]     Using only the color red, what is the greatest number of roads you can color in the six-city diagram without creating any red triangles? It's OK if some of the uncolored roads form triangles.

6. [Almost Impossible]     Suppose Black Parkways now enters the road-building game. If you erect 17 towns and connect them to one another with roads that can now be red, blue, or black, will you always form at least one triangle of a single color? Hint: Expand upon the solution to problem 3.

Solution

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