2014 Summer Wrangle problems
- 1. Week 1 Wrangle Problems 1. Is there an arrangement of the ten numbers 1 1 2 2 3 3 4 4 5 5 in a row so that each number, except the first and last, is the sum or difference of its two adjacent neighbors? 2. Nora wants her ark to sail along on an even keel. The ark is divided down the middle, and on each deck the animals on the left exactly balance those on the right–except for the third deck. Can you figure out how many seals are needed in place of the question mark so that they (and the bear) will exactly balance the six zebras? 3. The Locker Problem This is a famous brain-teaser. See if you can solve it! 100 students attend a school that has lockers in the hallway. On the last day of school, all of the lockers are unlocked and empty so the students decide to try an experiment. The first student opens all of the lockers. The second student closes the lockers numbered 2, 4, 6, 8, 10, and so on. The third student changes the lockers numbered 3, 6, 9, 12, 15, and so on. (The third student opens these lockers if they were closed and closes these lockers if they were open). The fourth student changes the lockers numbered 4, 8, 12, 16, 20, and so on. The process continues until the one-hundredth student changes the locker numbered 100. After their experiment, which lockers were open? What is special about these locker numbers? 4. Each face of a cube is divided into four squares. Each of the 24 squares is to be painted with one of three colors in such a way that any two squares with a common edge have different colors. Can you arrange to have nine squares of the same color? 1 of 3
- 2. 5. An ant starts in a corner of the floor of a cubical room (Al Ts´e). It wants to move to the opposite corner (Al Ts’´o) using the shortest route. It can only move along the walls, floor, and ceiling of the room. What path should it take? 6. At a big party, 101 people shook hands. Each person shook hands exactly once with each and every other person. How many handshakes, in total, were there? 7. Out of 9 identically-looking coins, one is counterfeit (a fake!), and it is lighter than the others. How can you discover the fake with only two weighings in a two-pan balance? 8. Find the least number of weighings on the balance necessary to discover one counterfeit among 27 coins. 2 of 3
- 3. Week 2 Wrangle Problems 1. The product of three consecutive integers is 8 times their sum. What is the sum of their squares? 2. (a) Arrange, if possible, 24 pennies flat on a table so that each one touches exactly 2 others. If not possible state why. Do the same thing with 25 pennies. (b) Now each touches 3 others. Solve for 24 pennies. Solve for 25. 3. (a) Everyday at school, Joe climbs a flight of 6 stairs. Joe can take stairs 1, 2, or 3 at a time. For example, Joe could climb 3, then 1, then 2 stairs. In how many ways can Joe climb the stairs? (b) Mary can take stairs 1 or 2 at a time and goes to the same school as Joe. In how many ways can Mary climb the same stairs as Joe? 4. The area of rectangle ABCD is 72. If the point A and the midpoints of BC and CD are joined to form a triangle, the area of that triangle is ...? 5. An 8 × 8 grid is covered by 1 × 2 dominoes. Prove that two of the dominoes form a 2 × 2 square. 6. If only two pieces of frybread fit in the frying pan at one time and it takes one minute to fry one side of each piece, what is the least time needed to fry both sides of three pieces of frybread? 7. At Middletown Elementary School, 4 9 of lower elementary students are female, and 2 3 of females are lower elementary students. Are there more females or lower elementary students at Middletown Elementary School? 8. A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games, and Lara won 2 games, how many games did Monica win? 3 of 3