MathCircle
MathCircle's cast carries the cooperative math-circle posture: a meta-host who steps back so kids talk to each other, a listener-restater who weaves social fabric, a wait-time keeper, and a dual-voice productive-failure surface.
Two ways to grow as a problem-solver
Browse the 60 circle problems
Cast circle
Warm-up Circle Circe draws a circle and places three dots on its edge, evenly spaced. She connects each dot to every other dot. How many distinct line segments appear inside (or on) the circle? Getting going Circle Circe draws a circle and places five dots on its edge, no three of which lie on a single line through the center. She connects every pair of dots with a line segment. How many distinct line segments appear? Getting going Five friends sit in a row. The first whispers '7' to the second, who multiplies by 2 and whispers the result to the next, who adds 3 and whispers it on, who divides by 2 and whispers it on, who subtracts 1. What does the last friend hear? Getting going Patty Patient watches the moon. It returns to the same phase every 29 days. If a new moon falls on day 1 of a year, on what day of the year does the next new moon fall? And how many new moons happen in a 365-day year if the first is on day 1? Getting going Tortoise wants to find 1 + 2 + 3 + ... + 20 by adding one number at a time. Hare wants to use a clever pairing trick. Find the sum, and show both ways. Which way feels right to you, and why might both ways belong in a math circle? Circle problem Circle Circe stands at the center of a regular hexagon with side length 6. She holds a lantern that lights every point within distance 6 of her. Does the lantern light every point inside the hexagon? Circle problem Echo Edie hears the following: 'Three friends share 12 marbles. The first gets twice as many as the second. The third gets the same as the first.' She restates: 'Let the second friend get m marbles. Then the first gets 2m and the third gets 2m. Total: 2m + m + 2m = 5m.' Use Edie's restating to find how many marbles each friend gets. Circle problem Echo Edie restates a problem: 'A pitcher holds some water. After pouring out one third, then one third of what's left, then one third of what's left again, two cups remain.' Edie writes: 'Let V be the starting volume. After the first pour, 2V/3 remains. After the second, (2/3)(2V/3) = 4V/9. After the third, (2/3)(4V/9) = 8V/27. That equals 2.' Find V. Circle problem Tortoise and Hare walk to a fountain along the same path. Tortoise walks 2 meters per minute the whole way. Hare walks 4 meters per minute but stops to nap for the second half of the trip and gets there just as Tortoise does. How long is the path to the fountain? Stretch Patty Patient watches a clock. The hour hand and the minute hand are together at 12:00. How many minutes pass until they are together again?
Zvonkin classics
Warm-up A traveler must take a wolf, a goat, and a cabbage across a river. The boat fits the traveler plus one item. If left alone together, the wolf eats the goat, and the goat eats the cabbage. How can all three be brought safely across? Warm-up How many triangles can you count in a figure made of an equilateral triangle subdivided into 4 smaller equilateral triangles by joining the midpoints of its sides? Getting going A chocolate bar is a 4 by 6 grid of squares, attached along the edges. How many straight breaks (along edges) are needed to separate all 24 squares? Getting going Fill a 3 by 3 grid with the numbers 1 through 9 (each used once) so that every row, every column, and both diagonals sum to the same value. What must that value be? Getting going If one person paints a fence in 6 hours, and another person paints the same fence in 3 hours, how long does it take them to paint the fence together? Getting going Ten coins are on a table, all heads-up. In one move, you may flip exactly two coins (any two, both at once). Can you reach a state where all ten coins show tails? Circle problem A frog sits on lily pad 1 of an 11-pad row. Each jump moves the frog forward by 1 or 2 pads. In how many different jump sequences can the frog reach pad 11? Circle problem Two trains start 200 km apart and travel toward each other on the same track, each at 50 km/h. A fly leaves the front of one train, flies at 75 km/h to the other train, turns around instantly, flies back, and continues until the trains meet. How far does the fly travel in total? Circle problem To multiply 27 by 41 using only doubling, halving (ignoring remainders), and addition: write 27 next to 41. Repeatedly halve 27 and double 41 until 27 reaches 1. Add up the doubled values whose halved value is odd. Show this gives the product 27 × 41. Circle problem Three pegs stand in a row. On the leftmost peg, four disks of increasing size are stacked, largest on the bottom. You may move one disk at a time from one peg to another, but never place a larger disk on a smaller one. What is the smallest number of moves to transfer the whole stack to the rightmost peg?
Berkeley Math Circle
Warm-up Eight friends meet at the circle. Each one shakes hands with every other person exactly once. How many handshakes happen in total? Warm-up A drawer holds black socks and white socks, mixed together in the dark. What is the smallest number of socks you must pull out — without looking — to be certain you have a matching pair? Getting going Find the sum 1 + 2 + 3 + ... + 100 without adding the numbers one at a time. Show why your method works. Getting going A wooden cube is painted on all 6 outer faces, then cut into 27 smaller cubes (3 cuts each direction). How many of the small cubes have paint on exactly two faces? Circle problem Two opposite corners are removed from a standard 8×8 chessboard, leaving 62 squares. Can the remaining board be perfectly covered with 31 dominoes, each covering two adjacent squares? Circle problem Four travelers must cross a bridge at night, sharing a single flashlight. The bridge holds at most two people at a time. The travelers walk at speeds of 1, 2, 5, and 10 minutes per crossing. When two cross together, they walk at the slower person's pace. What is the shortest total time for all four to cross? Circle problem What is the last digit of 7 raised to the 100th power? Circle problem On a 4 by 6 grid of city blocks, how many shortest paths go from the bottom-left corner to the top-right corner, moving only right or up along the streets? Circle problem You have a 3-liter jug and a 5-liter jug, both unmarked, and unlimited water. Show how to measure exactly 4 liters. Stretch There are 100 lights numbered 1 to 100, all off. Person 1 toggles every light. Person 2 toggles every 2nd light. Person 3 toggles every 3rd light. This continues through person 100. Which lights are on at the end? Stretch Five points are placed anywhere inside a unit square. Prove that two of them must be within distance √2 / 2 of each other. Stretch A knight starts on a corner of a standard 8x8 chessboard. What is the smallest number of knight's moves to reach the diagonally opposite corner? Stretch Two mirrors meet at right angles. A laser pointer aimed into the corner reflects off one mirror, then the other, and returns. Show that the returning beam is parallel to the incoming beam. Stretch A monkey climbs a rope that is exactly as long as the monkey's weight. The rope passes over a frictionless pulley to a basket of bananas of equal weight to the monkey. If the monkey climbs up the rope, what happens to the basket of bananas? Stretch Three identical-looking doors. Behind one is a prize; the other two have nothing. You pick door 1. A friend who knows which door has the prize opens door 3, revealing nothing. They offer to let you switch to door 2. Should you switch? Stretch An ant walks along the surface of a 1-by-1-by-1 cube from one corner to the diagonally opposite corner. What is the shortest such path's length? Stretch On a 4-by-4 board, place 4 knights so that no knight attacks another. How many ways are there (counting rotations and reflections as distinct)? Hard nut Twelve identical-looking balls — one is heavier or lighter than the rest. Using a two-pan balance and at most 3 weighings, identify the odd ball AND whether it is heavier or lighter. Hard nut Three friends agree to a hat game. Each will be given a red or blue hat (chosen by a coin flip for each). They will see the others' hats but not their own. At a signal, each must simultaneously guess their own color OR pass. They win if at least one guesses correctly AND no one guesses incorrectly. They cannot communicate after hats are placed. What strategy gives the highest chance of winning, and what is that chance? Hard nut On an infinite checkerboard, place checkers on every square below a horizontal line. Legal move: any checker jumps over an adjacent checker (horizontally or vertically) into an empty square beyond, and the jumped checker is removed. The goal is to land a checker as far above the line as possible. How high can you reach?
AMC-8 style
Warm-up The sum of three consecutive positive integers equals 36. What are the three integers? Warm-up A rectangular tank measures 4 ft long, 3 ft wide, and 2 ft deep. Water flows in at 1 cubic foot per minute. How many minutes until the tank is full? Getting going A jar holds only nickels (5¢) and dimes (10¢), totaling $2.40 from 30 coins. How many of each coin are in the jar? Getting going Two friends walk from opposite ends of a 1-mile path toward each other. One walks 3 mph and the other walks 5 mph. After how many minutes do they meet? Getting going A square has side 10. Inside it, a circle is inscribed so it touches all four sides. What is the area of the region inside the square but outside the circle? Getting going A street has 20 houses numbered 1 through 20. How many of the house numbers contain the digit 1 at least once? Getting going What is the smallest pair of consecutive positive integers that are both prime, OR explain why no such pair exists. Getting going A rectangle has perimeter 24 and integer side lengths. Among all such rectangles, which one has the greatest area? Circle problem At 3:15 on an analog clock, what is the smaller angle between the hour hand and the minute hand? Circle problem Six identical squares are arranged in a plus-shape: one center square, with one square attached to each of its four sides, plus one more square attached to one of those outer squares. Can this shape be folded along the edges to form the surface of a cube? Circle problem Find all two-digit positive integers that equal twice the sum of their digits. Circle problem How many three-digit positive integers (from 100 to 999) are divisible by 3 but not by 9? Circle problem A path winds up a mountain. Going up, walking speed is 2 mph. Coming down the same path, walking speed is 6 mph. What is the average speed for the round trip? Stretch Which positive integers cannot be written as a sum of two or more consecutive positive integers? Stretch Three standard six-sided dice are rolled. What is the probability that the three numbers shown can be arranged to form an arithmetic sequence with common difference 1? Stretch Using only 3¢ and 5¢ stamps (any number of each), which postage amounts cannot be made exactly? Stretch Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. Stretch How many trailing zeros are at the end of the number 25! (25 factorial)? Stretch A 5 by 5 grid is colored with two colors. Each row and each column must contain at least one square of each color. What is the smallest number of squares of the minority color needed? Hard nut A rectangle is folded so that one pair of opposite corners meet. The resulting crease has length 5. If the rectangle has width 3, what is its length?
