Puzzles

You are blindfolded and given a deck of 52 cards, exactly 24 of which are face-up. Can you partition the cards into two piles, each containing the same number of face-up cards?

Is it possible to partition a cube into two or more cubes with distinct sizes?

Can you hang a picture on two nails such that removing either one causes it to fall?

Given two points in the plane, construct their midpoint using only a compass.

Two hexagons share a border but do not overlap. What is the minimum possible number of sides in the resulting figure?

Is it possible to fit six \(1 \times 2 \times 2\) blocks into a \(3 \times 3 \times 3\) cube?

Using the numbers 3 3 7 13 each exactly once and some of the symbols + - * / ^ ( ), construct an expression that evaluates to 10.

Using the numbers 4 5 9 each exactly once and some of the symbols + - * / ^ . ( ), construct an expression that evaluates to 23.

Find a 10-digit integer \(n\) such that

• each digit appears exactly once in \(n\), and
• the integer formed by the first \(k\) digits of \(n\) is divisible by \(k\), for each \(k \in \{1, 2, \ldots, 10\}\).

A needle (a segment) lies on a plane. One can rotate it \(45^{\circ}\) around any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?

What is the maximum number of knights that can be placed on a chessboard such that each knight attacks exactly one other knight?

Shuffle a deck of cards, and turn cards over until you reach the first ace. Which of the following cards has a higher probability of being turned over next: the Ace of Spades or the Two of Hearts?

Four points form a square. You may repeatedly move any point to its reflection over any another point. Is it possible for the four points to ever form a larger square?

A triangle can be covered with 25 circles of radius 2. Prove that it can be covered with 100 circles of radius 1.

You and I are playing a game using a \(5 \times 7\) grid of chocolate. We alternate breaking any chunk of chocolate into two smaller chunks along the grid lines. If you start and the goal is to make the last move, who has the winning strategy?

Let \(p_1 < p_2 < \ldots\) be the primes in increasing order. Show that \(p_{n}+p_{n+1}\) always has at least six divisors for all \(n \geq 3\).

Reverse the digits of a nontrivial power of 7. Can the result ever be a power of 11?

What is the largest number of isosceles triangles that can be formed using six distinct points in the plane as vertices?

Six points are marked in the plane, with no three collinear. Consider the \(\binom{6}{2}\) lines formed by pairs of these points. What is the maximum possible number of triple intersections (not at marked points) among them?

A die is rolled until a 6 appears. Given that every number rolled was even, what is the expected number of rolls?

A parabola and a circle intersect at exactly two points \(A\) and \(B\). Suppose that the circle is tangent to the parabola at \(A\). Does it follow that the circle is tangent to the parabola at \(B\) as well?

Does there exist a two-variable polynomial \(P : \mathbb{R}^2 \to \mathbb{R}\) whose image is \(\mathbb{R}^+\)?

Two parallel line segments are drawn in the plane. Using straightedge alone, divide one of them into six congruent parts.

The cost of mailing a rectangular parcel is proportional to the sum of its three dimensions. Is it possible to save money by shipping a parcel inside another parcel?

Cevians \(\overline{BE}\) and \(\overline{CF}\) of triangle \(ABC\) meet at \(P\). Prove that \(AE+EP=AF+FP\) if and only if \(AB+BP=AC+CP\).

The vertices of a quadrilateral in \(\mathbb{R}^3\) are tangent to a sphere at four points. Prove that these four points are coplanar.

The numbers \(1, 2, \ldots, 9\) are written on nine tags. You and I alternate choosing tags, and the goal is to be the first to obtain three tags that sum to 15. If you go first, do you have a winning strategy?

Does there exist a function \(f : \mathbb{R}^2 \to \mathbb{Z}\) for which \(f(x, y) = f(y, z)\) implies \(x = y = z\)?

Is it possible to fold a square piece of paper to produce a flat figure that has a larger perimeter than the original square?

Is it possible to color each positive integer either red or blue such that no infinite arithmetic progression is all red or all blue?

I choose two distinct real numbers and seal each one in its own envelope. You choose one of the envelopes, open it, and guess whether or not the unopened envelope contains the larger number. Is there a strategy where you guess correctly over half the time, regardless of what numbers I choose?

A Y is a subset the union of three finite paths joined at a common endpoint. Is it possible to draw uncountably many disjoint Y's in the plane?

There are 1000 cows. Prove that one cow can be removed so that the remaining 999 cows cannot be split into two groups with equal weights.

The angles of a hexagon in \(\mathbb{R}^3\) are all right, and no pair of opposite sides are parallel. Prove that there exists a line perpendicular to the three lines perpendicular to each pair of opposite sides.

Can \(\mathbb{R}^3\) be partitioned into disjoint nontrivial circles?

Index the set of all finite Turing machines with positive integers and let \(K\) be the set of all \(n\) for which the \(n^\text{th}\) Turing machine halts. You are given a set \(\{n_1, n_2, n_3\} \subset \mathbb{Z}^+\), and your task is to determine \(\{n_1, n_2, n_3\} \cap K\). If an oracle can tell you whether or not \(n \in K\) for at most two values of \(n\), is it always possible to complete your task in finite time?