The Great Floobini
The great Floobini has developed a new magic trick. In a black velvet bag, Floobini has eight stones, each engraved with a different number. A volunteer from the audience reaches into the bag, picks out some stones, and tells Floobini the sum of the numbers on the stones. Floobini then amazes his audience by revealing the numbers on each stone. This isn't an easy trick. Floobini first tried it with stones numbered 1, 2, 3, 4, 5, 6, 7, and 8, but those numbers often add up to the same sum. For instance, 1, 2, 3, 4, and 5 add up to 15, as do 7 and 8. Then he tried the numbers 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, and 10,000,000. This was easier but too obvious, and the audience was not impressed. See if you can match Floobini's brilliance:
1. What are the eight smallest whole numbers that will make Floobini's trick work? Make the smallest number 1 and the largest number as small as possible.
2. If Floobini asks his volunteer to pick only two stones from the bag and tell him the sum of their numbers, what are the eight smallest numbers that will work? Again, the largest number should be as small as possible. What are the eight smallest numbers that will work if three stones are chosen?
3. Floobini's lazy rival, Gloobini, has stolen Floobini's trick. Gloobini added 1 to each of Floobini's numbers in the first part of problem #2. Although the trick works, the largest number in the set has increased. Is there another set of eight numbers in which the largest number in the set is the same as the one in the answer to problem #2?
The eight checkers on the 4x4 checkerboard to the right form the corners of three squares: a small square and two larger, tilted squares.
1. If you place checkers on all 16 squares of a 4x4 checkerboard, how many squares do they form? Count any square that has a checker at each corner. Squares can be any size, straight or tilted.
2. What is the largest number of checkers you can place on a 4x4 checkerboard without forming any squares? What is the largest number of checkers that don't form any squares on a 6x6 checkerboard? And the hardest challenge: on an 8x8 checkerboard? Remember that the squares can be any size, straight or tilted.
A Perfect Match
The figure below shows an L-shaped area outlined with matchsticks. The area of the shape is three squares, and there are eight matchsticks.
1. Using the matchsticks as the edges of a grid, find a shape that uses the same number of matchsticks as its area. What is the smallest number of matchsticks you can use?
2. What other number of matchsticks on a grid can outline an area that is equal to the number of matchsticks used?
3. Just as a square's area can equal its perimeter, a circle's area can equal its circumference. Can you figure out the diameter of the smallest circle whose area equals its circumference?
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