Math Circle Problems
Traveling on the edge of a cube
Is it possible for an ant to begin at one vertex (corner) of a cube, crawl along all the edges, and then return to its starting point without retracing any portion of its path?
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Three Cent Coin
Suppose that a certain country has a 3-cent coin and a 4-cent coin. Then it is
not possible to obtain certain amounts of money using only these coins, such as
five cents. Determine which amounts can be obtained with these coins.
Now repeat the previous problem with a 3-cent and 5-cent coin. Then try it once more with a 3-cent and 7-cent coin.
Based on the results of the previous two problems, make a conjecture (an
educated guess) as to the largest amount that cannot be obtained using only
3-cent and b-cent coins for any value of b not divisible by 3. Also conjecture
the number of amounts that cannot be obtained.
From Circle in a Box.
Friday the 8th
What are the maximum number of Friday the 8ths that can occur in a normal
365-day year? What are the minimum number that must occur?
Hint: (Recall that April, June, September and November each have 30 days, February has 28 days, and all the other months have 31 days.)
Completing the Square
Choose any positive integer and compute its square. Then add both your original
number and the next higher integer to this square. The result will be the next
perfect square! Use a picture to explain why this trick works.
A Ladder Locus
A ladder, originally upright against a vertical wall, slides downward and sideways, always in contact with the wall,
until it crashes to the ground. What figure is traced out by a painter's
paintbrush, which is attached midway up the ladder, as it slides down the wall?
Further Exploration: How does the figure change if you move the paintbrush closer to the top or nearer to the bottom of the ladder?
From Circle in a Box.
Squares in a Square
How many squares can you count?
Need a Hint?
Challenge Puzzle: For the same grid of sixteen squares how many
rectangles can be formed?
(remember that squares are rectangles)
Click Here for the solution
From MSRI Puzzles on Wheels Archive.
Puzzle on Wheels: Tracing a Route
Which of these designs can you draw without lifting your pencil from the paper (drawing each line once and not drawing any extra lines)?
Need a Hint?
Challenge Puzzle:
It is impossible to draw the following design without lifting your pencil from the page, traversing each and every edge precisely once?
Suppose now, however, you are permitted to trace over a small number of edges twice in order to complete the picture. What is the minimal number of edges that must be reused to draw the figure?
Click Here for the solution.
From MSRI Puzzles on Wheels Archive.
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