My Third Solution to the Zome Cryptocube Puzzle

The President of the Zometool Corporation, Carlos Neumann, gave me a challenge, not long ago: find a solution to the Zome Cryptocube puzzle which uses only B0s, which I call “tiny blue struts.” For the Cryptocube puzzle, though, these “blue” struts actually appear white. Carlos knows me well, and knows I cannot resist a challenge involving Zome. Here is what I came up with, before the removal of the black cube, which is what the Zome Cryptocube puzzle starts with.

150923_0000

In a “pure” Crypocube solution, the red Zomeballs would also be white — not just the “blue” struts. However, when Carlos issued this challenge, I was at home, with all the white Zomeballs I own located at the school where I teach — so I used red Zomeballs, instead, since I had them at home, and did not wish to wait.

Here’s what this Cryptocube solution looks like, without the black cube’s black struts. You can still “see” the black cube, though, for the black Zomeballs which are the eight corners of the black cube are still present. As is happens, this particular Cryptocube solution has pyritohedral symmetry — better known as the symmetry of a standard volleyball.

150923_0001

While the Cryptocube puzzle is not currently available on the Zome website, http://www.zometool.com, it should be there soon — hopefully, in time for this excellent Zome kit to be bought as a Christmas present. Once a child is old enough so that small parts present no choking hazard, that child is old enough to start playing with Zome — and it is my firm belief that such play stimulates the intellectual growth of both children and adults. As far as a maximum age where Zome is an appropriate Christmas gift, the answer to that is simple: there isn’t one.

Also: while I do openly advertise Zome, I do not get paid to do so. I do this unpaid advertising for one reason: I firmly believe that Zome is a fantastic product, especially for those interested in mathematics, or for those who wish to develop an interest in mathematics — especially geometry. Also, Zome is fun!

Geometry Problem Involving Two Circles (See Comments for Solution)

This is a puzzle I made up not long ago. After trying to solve it for a bit (no success yet, but I haven’t given up), I decided to share the fun.

A small circle of radius r is centered on a large circle of radius R. It is a given that 0 < r < R. In terms of r and R, what fraction of the smaller circle’s circumference lies outside the larger circle?

two circles

I am 90% certain there is an extremely simple way to do this, using only things I already know. It’s frustrating that the answer isn’t simply leaping out of the computer screen, at me. For simple math problems, that’s what usually happens . . . so either this is merely deceptively simple, or I am missing something.

Silly U.S. Map Puzzle #5

What do the colors on this map mean?

mappuzzle5

If you wish to check your answer, or just what to know what the solution is, just scroll down.

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And keep on scrollin’. . . .

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Solution:

Of the other 49 states in the USA, how many are adjacent to this one? The answer to this question determines the color of each state.

One point of clarification: if it takes a lengthy trip by boat or ship to get there, I didn’t count it as an adjacent state . . . so, for example, Minnesota and Michigan didn’t make each other’s lists. Simply going over a bridge isn’t enough for this sort of separation, though, which is why Arkansas and Tennessee did make each others’ lists of adjacent states. Had I interpreted water borders differently, this map would have some differences.

Another way this map could be altered would be to count states that meet others only at a single point, rather than a border with non-zero length. This would change the colors of the “four corners” states of Arizona, Utah, New Mexico, and Colorado, but would have no effect on the other 46 states.

Silly U.S. Map Puzzles #4a and 4b

First, for puzzle #4a, what are the meanings of the colors on this map?

mapquiz4a-letters

For puzzle #4b, what do the colors mean on this second, similar map?

mapquiz4b-characters

To find the answers, simply scroll down.

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Solution:

In the first map, consider the number of letters in the name of each state. Is this number prime or composite?

In the second map, consider the number of characters, rather than letters, in each state’s name. This number is different for states with two-word names, due to the single character, a blank space, needed to separate the two words. Again: prime, or composite?

Silly U.S. Map Puzzle #3

What is represented by the colors on this map?

mapquiz3

The answer may be found by scrolling down.

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Answer:

Do any of the borders of this state contain squiggles? (Note: if you think New Mexico is the wrong color, check the part of that state which borders El Paso, Texas.)

Silly U.S. Map Puzzle #2

What is represented by the colors on this map?

mapquiz2

If you give up, you can scroll down to find the answer.

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Answer: the colors show whether the name of each state starts with a letter in the first, or second, half of the alphabet.

Silly U.S. Map Puzzle #1

What is represented by the colors on this map?

mapquiz1

If you decide to give up, you can scroll down for the answer . . . but, I promise, the solution to this puzzle is extremely simple.

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Keep scrolling, if you’re looking for the answer….

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Answer:

The map shows how many words are in the name of each state.