Six words are to be entered, one across each row of the grid.

The colors are present to create individual regions. In each region all the letters are the same. Two adjacent regions have different letters; non-adjacent regions may or may not have the same letter regardless of same or different color.

Clues for the words in the four middle rows (rows 2-5), in random order, are:

- hot dog topping
- VCR's "go back to start" button
- negligent
- language of Warsaw

From Mensa Puzzle Calendar 2019 by Fraser Simpson, Workman Publishing, New York. Puzzle for June 14.

Take a 7x7 grid of squares and remove the center square.

Take two 5x5 grids of squares and remove each center square.

Cut the smaller figures into as few pieces as possible and reassemble to form the larger.

Is it possible to use fewer pieces if you are allowed to remove a different square than the center any of the original shapes?

Lucy stands on a set of faulty scales that state her mass as 48 kg. David's mass from the same scales is recorded as 56 kg and their combined mass is recorded as 107 kg.

Assuming the magnitude and nature of the errors was constant, what is the true mass of David?

Let f:ℝ→ℝ be twice differentiable such that f(0)=2, f'(0)=-2, f(1)=1

Find for at-least how many c ∈ (0, 1)

f(c).f'(c)+f''(c)=0

is satisfied.

There are two identical uniform spherical planets of radius R. The first has its center at the origin of the xyz coordinate system. The second has its center at (2R, 0, 0). The planets are touching.

A projectile is launched from the "North Pole" of the first planet at (0, 0, R) with its initial velocity pointed in the direction of the vector (1, 0, 1).

Let the escape speed relative to the planet's surface be v_{e}. Note that here, the escape escape is for a single planet in isolation (following the typical convention).

With the given launch vector, let v_{0} be the minimum launch speed for the projectile to reach the "North Pole" of the second planet at (2R, 0, R).

How are the two speeds v_{e} and v_{0} related?

If cos x is irrational, find maximum positive integer n such that cos 2x, cos 3x, ... cos nx are all rational.