21-30

21. This was a recent puzzler from CarTalk.
I noticed that the last 4 digits were palindromic, that is they read the same forwards as backwards. One mile later, the last 5 numbers were palindromic. One mile after that, the middle 4 out of 6 numbers were palindromic. And you ready for this? One mile later, all 6 were palindromic!The question is, what did I see on the odometer when he first looked? (1 Point)

22. The numbers 1-3 can easily be configured into a "difference triangle", an inverted triangle so that a lower number is the absolute difference between the two numbers above it. For exampleare all solutions. Find a three-row difference triangle for the numbers 1-6 using each number exactly once(1 Point), a four-row difference triangle with the numbers 1-10 using each number exactly once(2 Points), or a five-row difference triangle with the numbers 1-15 using each number exactly once (3 Points).

23. Prove that there is no six-row difference triangle (defined in 22.) using the numbers 1-21 using each number exactly once. (5 Points)

24. Given a triangle, construct a triangle with twice the area using only a straight edge and compass. (1 Point)

25. A child is given the task of painting a box full of identical cubic wooden blocks with two colors – red and blue. Each face is to be either all blue or all red, never both blue and red on the same face. The blocks can be colored with all blue faces, all red faces or some combination of blue faces and red faces. How many different kinds of colored blocks the child can make? (1 Point)

Note: Two blocks are same if they can be put into matching positions by rotating. For example, all blocks with five red faces and one blue face are same.

26. A dart thrower hits any point on the dart board with equal probability. His dart board has an inner circle with half the radius of the whole board. What is the likelihood that he hits the inner circle in a single throw? (1 Point)

27. Construct a regular hexagon with only straight edge and a compass. A regular polygon is one with all edges the same length and all angles equal. (1 Point)

28. A golden rectangle is a rectangle whose proportions remain the same after removing the largest possible square with a single cut. Construct a golden rectangle with only a straight edge and compass. (2 Points)

29. Construct a regular pentagon with only a straight edge and a ru. (3 Points)


30. What is the value of x when

What is the value of x when

Explain the similarities or differences between the real value of x for the above two equations. (2 points)

Next 10

11. How many ways can you arrange 12 (1x2) dominoes into a 12x2 rectangle? (1 Point)

12. The area of the small square inscribed in the circle is one square inch. What is the area of the larger square? (2 Points)

13. How many ways can you make the sum of 11 using only the numbers 1 and 2? Count each ordering separately, that is
1 + 2 + 2 + 2 + 2 + 2 and
2 + 1 + 2 + 2 + 2 + 2 count as two ways. (1 Point)

14. There are three pairs of colored balls: two green, two red, and two blue. One ball from each pair is defective (weighs less). All three defective balls are of the same weight, so are the three perfect balls. Use two weighings on a simple balance scale to determine which balls are light and which are perfect (3 Point).

15. The area of the small hexagon inscribed in the circle is six square inches. What is the area of the larger hexagon? (2 Points)

16. Connect nine points arranged in a 3x3 square using 4 straight line segments without lifting your pencil or backtracking. Recall points have zero width they have been enlarged for clarity. (1 Point)

17. How many ways can you arrange five black billiard balls and five white billiard balls in a line? For instance "WWWWWBBBBB" is one such arrangement. All billiard balls must be used and spaces are not allowed. (2 Points)

18. How many combinations are there on a five button cypher-lock? On these locks you can press 1, 2, 3, 4, or 5 buttons at a time so long as each button is pressed at most once. For example pressing buttons 1 and 2 simultaneously followed by 4 and 5 simultaneously is a valid combination, whereas pressing buttons 1 and 2 followed by 2 and 3 is not. (3 points)

19. Solve the Rubik's Cube without help. (6 points)

20. A certain species of amoeba splits every minute. An hour after a pair of amoeba are put into the jar, the jar is full of amoeba. How long does it take for a single amoeba to fill the jar? (1 Point)

First 10

1. Find the length of the hypotenuse of a the triangle. (1 Point)

2. How many 10-digit numbers can be written by using all 10 digits 0-9. (Numbers starting with 0 do not count) (1 Point)

3. Of 9 coins of the same denomination, 8 weight the same, and one, a counterfeit, is lighter than the rest. Find the counterfeit using a balance only two times without weights. (1 Point)

4. Same as three only the counterfeit may be light or heavy, there are 12 coins, and you may use the balance three times. (3 Points)

5. How far does a ball travel if it is dropped from the Leaning Tower of Pisa (179 Feet) if it rebounds 10% of its height after each bounce. (1 Point for nearest foot, 2 points exact)

6. Express 100 in three different ways with five 5s. You may use these symbols + - / * ( ). (1 Point)

7. 2+2 = 2*2, 1+2+3 = 1*2*3. Find four positive integers whose sum equals their product. (1 Point)

8. Same as seven with 5 numbers. (1 Point)

9. There are 100 lockers in a row that are all closed at the beginning. There are 100 students. Each student makes a pass. The first student opens the every locker door. The second student toggles (if the door is closed, you open it, if its open, you close it) every 2nd locker door (#2, #4, #6 etc) starting with locker #2. The third student toggles every 3rd locker door starting with locker #3. The n^th student toggles every n^th locker door starting with n^th locker.
Which lockers will be open at the end? (1 Point)

10. A mathematician attends a dinner party with his wife and four additional couples. When the guest arrive various hand shakes take place with no one shaking their own hand nor that of their spouse. When they sit down for dinner the mathematician asks each of the others (including his wife) how many hands they shook. To his surprise, he found that each had shook a different number of hands! How many hands did his wife shake? (1 Point)