Calendar

Here is a synopsis of the 2007--2008 year. The Circle met from 4:00 p.m. to 5:30 p.m. on Mondays at the Carriage House Conference Center, behind the headquarters of the Mathematical Association of America near Dupont Circle. The address is 1781 Church Street, NW, Washington, DC 20036.
September 24 - We played take-away games with tokens. Suppose that a pile of tokens (we used "pogs") is placed between two players, who alternately remove a number of tokens, where that number is restricted to a certain set of positive integers. For example the set might be {1,2,3,4} and the original pile might contain 2007 tokens. How should the players proceed? Does the first player have a winning strategy? What if the set is changed to {1,2,3}? To {1,2,4}? To {1,2,5}? To {1,2,6}? Circle participants discovered how to play most of these simple subtraction games, and they produced explanations of why their strategies worked. At the end of Circle time, we introduced the game of Nim.
October 1 - We constructed a piece of Pascal's Triangle and looked for patterns. What are the row sums? What are the sums of squares of the elements in a row? What are the alternating sums of rows? What are various diagonal sums? Where are the even and odd numbers? We produced an argument that the row sums are powers of 2. At the end of Circle time, we counted 10-bit strings with exactly 5 zeroes and 5 ones. We conjectured that questions of this type are answered by the elements of Pascal's Triangle and that the answer to this particular question is 252, the central element in the 11th row of Pascal's Triangle.
October 8 - This was a Federal Holiday, and the Circle did not meet.
October 15 - We built polyhedra out of toys called "Polydrons". We assembled the five Platonic solids, and discovered that there were no more regular polyhedra. But we also discovered a number of semi-regular polyhedra (Archimedean solids), and we assembled a truncated icosahedron, a rhombicosidodecahedron, and a rhombicuboctahedron. For each of these solids, we counted vertices, edges, and faces, and it was natural to conjecture Euler's formula: V+F=E+2.
October 22 - We drew graphs, and asked if the drawings were traceable without lifting the pencil from the paper. For some graphs, the answer was yes, but for others it was no. Participants discovered that it is often important to start at the right vertex when attempting to trace a graph. One student conjectured that more than two vertices of odd degree makes such a tracing impossible. We saw why this was true, but that leaves the question of whether it is possible to trace any graph with two or fewer vertices of odd degree. At the end of Circle time, we drew graphs with no crossing edges and counted vertices, edges, and faces. Euler's formula again!
October 29 - We considered the average and sum of the numbers from 1 to n and of the numbers from m to n. Participants derived the well known formulas. We remarked on the fact that there are n-m+1 numbers from m to n. The "+1" there is easy to overlook. We then considered the following puzzle: Which positive integers can be expressed as the sum of two or more consecutive positive integers? The number 14 can, for example, since 14=2+3+4+5. Participants experimented and came out with a series of conjectures that were refuted with examples. Finally a conjecture came out that nobody could refute: A number can be so expressed if and only if it is not a power of 2. One participant asked if we permit zero to be part of the sum. "Sure, but does that change anything?" "Yes, because then we can express 1 (which IS a power of 2) as 0+1, destroying the conjecture." Interesting.
November 5 - We counted squares (of all sizes) in an n-by-n chessboard. We also counted the number of dots in a triangular arrangement of dots beginning with n dots in the bottom row, n-1 dots above them, and so forth down to 1 dot at the top. Participants noted that the first count was given by the sum of the first n square numbers and that the second was given by the sum of the first n positive integers. The latter was quickly computed in closed form. The sum of two consecutive triangle numbers was seen to be a square, and this observation was verified via algebra. Students were then asked to count the number of pairs that could be chosen from 5 objects. The answer: again a triangular number! Something to investigate in future sessions.
November 12 - This was a Federal Holiday, and the Circle did not meet.
November 19 - Neither did we meet on this day, as the Thanksgiving holiday approached.
November 26 - Participants enumerated the shortest ways to walk from 17th and K Streets to 21st and G Streets in DC. That's 4 blocks west and 3 blocks south. We then discussed how many ways the letters in "EAT" can be arranged. Participants responded with "6", although there was some discussion about whether the arrangement had to be an English word. Partipants then discovered that there are 6! arrangements of the letters in "MONDAY" (one of them is "DYNAMO"), that the number of arrangements of the letters in "CIRCLE" is 6!/2=360 (note the serendipity there), and that the number of arrangements of the letters in "MISSISSIPPI" is 11!/4!4!2!1!. Finally students enumerated the arrangements of the letters in "SSSWWWW", and we had our answer to the original problem of walking to 21st and G. Pascal's triangle made a surprising appearance.
December 3 - We played paper and pencil games, like sprouts, discovering that this apparently strategic game is not strategic after all. We then explored the winning strategy for the game of nim. Amazing!
December 10 - We warmed up by counting the number of n-bit strings with no consecutive 1's. We then tiled an 8-by-8 chessboard with 32 dominoes, but we couldn't do it 31 dominoes if the original chessboard had its opposite corner squares removed. We saw why. We tried to tile a 6-by-6 chessboard with 18 dominoes while avoiding "fault lines", but again we couldn't do it. We saw why. There was no point in attempting to tile an 8-by-8 chessboard with 3-by-1 rectangles, because 64 is not divisible by 3. But it is easy to tile an 8-by-8 chessboard with 4-by-1 rectangles. What about tiling a 10-by-10 chessboard with 25 4-by-1 rectangles? Seems possible, but isn't. We found an explanation for this.
December 17 - This week we explored cellular automata. We began by playing with one-dimensional cellular automata on the web. We created a number of transition rules and watched to see whether the resulting automaton was periodic or apparently chaotic. Some patterns looked like Pascal's binary triangle. We then played with Conway's game of life on the web. We were able again to create our own transition rules, and we saw amazing patterns. The interesting rules require a delicate balance between birth and death of cells; otherwise the population grows without control or shrinks rapidly to nothing.
December 24 - Christmas eve. The Circle did not meet.
December 31 - New year's eve. The Circle did not meet.
January 7 - One more week off.
January 14 - We return to start off the new year by counting and summing divisors. How many divisors does 1,000,000 have? How many divisors does 10! have? What is the sum of the divisors of these numbers? Is there a number that equals the sum of its proper divisors? Yes, 6 is such a number, because its proper divisors are 1, 2, and 3, which do indeed sum to 6. Perfect! Can you find any other such numbers?
January 21 - Martin Luther King Day, a Circular holiday.
January 28 - We listed the squares of the first 21 nonnegative integers. What numbers can be expressed as the difference of two squares? What numbers can be expressed as the sum of two squares? What numbers can be expressed as a^2+b^2-c^2? Can 0 be expressed nontrivially in that form? What numbers can be expressed as the sum of three squares? The sum of four squares?
February 4 - Camille McKayle introduced the four-numbers game. Write four numbers on the vertices of a square. Then on the midpoint of each side of the square write the absolute value of the difference between the numbers on the two incident vertices. Then draw a square that connects the four new numbers, and do the process again, writing new numbers on the midpoints of the sides of the new square. Continue repeating this process. What happens? How soon? Does it matter what kind of numbers we use originally (whole numbers, fractions, real numbers)? What about the similar game played with a triangle, pentagon, or hexagon replacing the original square?
February 11 - We played with a large collection of toys and puzzles. A number of these toys involved separating metal pieces or removing metal rings. Others involved packing objects, arranging tiles in the plane, or assembling pieces to form a 3-dimensional shape.
February 18 - President's Day holiday.
February 25 - We introduced the AMC-10 contest via a team format. We split the group into two teams and gave the teams 15 minutes to prepare. We then called the participants up to the board in random order to present one of the first n problems on the 2004 AMC-10. Each of the n participants could choose any problem that was not already chosen. Teams cheered for their teammates and questioned the solutions of their opponents.
March 3 - Ivars Peterson presented a special session on dice. Dice of all kinds. Big dice, small dice, Casino dice, polyhedral dice, Sicherman dice, weighted dice, nontransitive dice, binary dice. We played some dice games. We even tried to invent some games.
March 10 - We began studying the positive integers n with the property that 1/n is a terminating decimal. Which n have this property? We conjectured and then proved the answer. We then turned to studying divisibility, starting by reviewing the special rules for testing divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11. Why do these rules work? Is there a divisibility rule for testing divisibility by 7? We mentioned the prime factorization of integers and saw how this gives an easy way of testing whether one number divides another. Is 30! a multiple of 1,000,000?
March 17 - We played some mathematical games, including SET, Blokus, and Othello. We discussed some mathematical questions that one can ask about such games. How many cards can be dealt in SET without a set appearing? (This is a difficult question!)
March 24 - Spring break. The Circle did not meet.
March 31 - What is the smallest number with exactly 100 divisors? It may be challenging enough just to name a number with exactly 100 divisors. One example is the number 2^99, 2 to the 99th power. (What are its divisors?) But this isn't the smallest such number. We saw that the additive structure of the natural numbers is simple (all numbers have the form 1+1+1+1+...+1), while the multiplicative structure of the natural numbers is complex (there are infinitely many primes).
April 7 - We explored the On-Line Encyclopedia of Integer Sequences and learned how to use it to find the next term of a sequence. For example, we discovered that the number of squares found on an n-by-n chessboard is 1, 5, 14, 30, and 55 for n=1, 2, 3, 4, and 5, respectively. Finding the sequence at the Encyclopedia website showed us the formula and told us dozens of other facts about this sequence. We found the sequence of primes, the sequence of Fibonacci numbers, and the sequence of squares. We found a number of other interesting sequences while looking around, such as the sequence of Chen primes or the sequence of Iccanobif numbers.
April 14 - We examined some tiling problems. We asked how many dominos it takes to cover a chessboard. We then asked if 31 dominoes could cover a chessboard missing the two squares at the ends of a long diagonal. We noticed that there was no domino tiling of a 6-by-6 square without a "fault line". We explained this via a counting argument.
April 21 - We examined several look-and-see proofs of famous mathematical assertions, like the formulas for the difference between two squares, for the sum of the first n positive integers, and for the sum of the squares of the first n positive integers.
April 28 - We discovered that all Pythagorean triples seem to come from an amazing formula.
May 5 - We contemplated the Principle of Mathematical Induction and learned how to use it to prove statements about whole numbers. We looked at the sum of the first n positive integers and saw how to prove via mathematical induction that it equals n(n+1)/2 for all n. We also considered the proof by mathematical induction that any 2^n-by-2^n chessboard missing one square can be tiled by L-shaped triominoes.
May 12 - We played two-by-two zero-sum matrix games, and we discussed optimal strategies for them. We contemplated non-zero-sum games as well, including the prisoner's dilemma.
May 19 - We discussed a variety of theorems about the concurrence of cevians in triangles.
May 26 - Memorial Day. The Circle did not meet.
June 2 - Our final session for the 2007--2008 year, with a party. We played DC Math Circle Jeopardy.