Family Math

Man Wrinkle spent one-fourth of his life as a boy, one-eighth as a youth, and one-half as an active man. If Man Wrinkle spent 10 years as an old man, then how many years did he spend as an active man?

A man walks one mile south, one mile west and then one mile north, ending where he began.  Where did he start the journey (there are actually many solutions, can you find more than one)

Imagine that you can use only the numbers 1 and 2.  In how many ways can you write any given numbers using only these numbers as the following examples show:

1=1  (1 way)
2=(1+1) = 2  (2 ways)
3=(1+1+1)=(1+2) = (2+1) (3 ways)
4= (1+1+1+1) = (1+1+2) = (1 + 2 + 1)= (2+1+1) = (2+2) (Surprise,  5 ways)
5 -->  8 ways
6 --> 13 ways
7 --> x ways.  What is x?

If you figured out the answer and want to share it please fill the following survey.

Link to Survey

There is one letter in error in the addition below.  How should this problem read in order to be solvable?  Except for this single wrong letter, the usual rules of such "encrypted" math problems apply:

• the same letters stand for the same digits wherever they appear
• different letters stand for different digits
• no leading zeros.

In this week's puzzle, we will consider a mathematical problem that is just one illustration of a much bigger theme in mathematics:  Do we have enough information to solve the problem?

Let me get right to the puzzle first:

It doesn't look too hard but when you start to write it out, you may find your self with too many unknowns.  This sort of problem is called a Diophantine equation.  You will have to finesse a bit of implied information out of the restriction that only whole number solutions are permitted.  You would never buy half of a mouse, now would you?

I will post a link to the source of this problem next week because in that same page is the answer and an excellent discussion of the the solution technique and of Diophantine equations in general.

Have fun. 
-Prometheus

Greetings from Prometheus, your host on the Family Math puzzle page.

Though I have not yet materialized in Tompkins County, I should be able to place a weekly puzzle on this page and follow it in a week with a solution.  I will start by fetching puzzles from some of the many excellent puzzle sites that are on the web.  When I get handy with the tools for taking comments and responding, we can begin discussing strategies for problem solving, making up puzzles by request and just generally going where ever our curiosity takes us.

Today's puzzle comes from a particularly steady and respected source, the Middle School Problem of the Week produced by the Latin School of Chicago.
Perhaps they should try hosting a MATH Olympics?

The Water Jug Problem
Imagine that you have at your disposal a supply of water, a 5-gallon container, and a 3-gallon container.  The containers have no markings on them. Your job is to come up with exactly four gallons of water using just the two containers. Write how this could be done.

Interactive Scratch Version Programmed by Jasper

A cook with a large knife has a cube of cheese 3 inches on a side. She wishes to slice cube into 27 small cubes 1 inch on a side. How can she do this with only six cuts?

What is amazing about this puzzle is that you can actually solve it by doing it yourself and you can use an apple instead of cheeze also, bu the cubes will be different (except the middle one!).  The best way to solve puzzle is to kind of "get into them".
Once you solved it you can come up with other variation of the puzzle.

The original puzzle has a twist to it:

"Why isn't it possible to cut the cheese in less than six cuts even if she can rearrange the pieces after each cut?"  (hint:  How many faces does a cube have?)

Please fill the survey (parent and/or child)

A snail creeps 3 ft. up a wall during the daytime. After all the labor it does throughout the day, it stops to rest a while... but falls asleep!! The next morning it wakes up and discovers that it has slipped down 1 ft while sleeping. If this happens every day, how many days will the snail take to reach the top of a wall 15 ft in height?

If 3 cats catch 3 rats in 3 minutes, how many cats will catch 100 rats in 100 minutes?

Try not to read others solutions but after you think on your own...

1. Three kinds of apples are mixed in a basket. How many apples do you need to take to make sure you have at least two apples of the same kind, (that means at least one pair of similar apples) ? All together there are 24 apples in the basket, 8 from each kind.