Best of the MathNerds

These are some of the best questions (and answers) that we have received at MathNerds. Each item has the original question, followed by a sequence of hints, and ending with a complete solution.

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New problem posted November 2002

• The Twelve Coins (or Twelve Bags of Gold) Find the false coin in three weighings of a balance scale, when you don't know whether the false coin is heavy or light.

New problems or solutions posted October 2002

• Buildings and Ladders How high do the ladders reach? An exercise from the Field Team.

• The Magnificent Seven Writing 1 as a sum of seven reciprocals.

• An Excess of Divisors Why are there more divisors ending in 1 or 9 than in 3 or 7?

• Mazurkiewicz's Theorem on 2-Sets Can you construct a set of points in the plane such that every line of the plane intersects the set in exactly two points?

• The Mirror Trick How to maximize cross-sections without calculus.

• Non-Dividing Sets What's the biggest subset such that no integer divides any other?

New problems or solutions posted April 2002

• Factor This Factor the expression A³(B-C) + B³(C-A) + C³(A-B).

• The Hole Problem A napkin-ring invariant.

New problems or solutions posted March 2002

• Digit Reversal Solve the equation 4 * (ABCDE) = EDCBA.

• Drawing the Line Given a finite set of points in the plane, is there a line intersecting exactly two of them?

New problems or solutions posted February 2002

• The Counterfeit Penny Find the counterfeit using a balance scale.

• Rock Star Shuffle Get the band U2 across the bridge in 17 minutes.

Problems with hint sequence and complete solutions

• The first digit of Which digit occurs most frequently?

• The integral challenge Evaluate an integral without integrating.

• The shortest road network Building the shortest road connecting four towns.

• Crazy Dice Our Crazy Dice work just as well as old boring regular dice and are more fun.

• Irrational Exuberance Is the sum of this infinite series irrational?

• The Notorious Jumping Function of Continuum County How discontinuous can a monotonic function be?

• The Nasties of the Round Table Math as a corporate survival strategy.

• Pell Mell We're looking for a few good square roots.

• The Buckled Rail Who would notice a error of one foot in a mile-long rail?

• Chipping N Stacking poker chips for fun and profit.

• McNuggets Building your McNugget collection.

• The Case of the Missing Mangos Fractional ownership in a mango cooperative.