An Ant’s Path

An ant is positioned at one of the vertices of a cube and wants to get to the opposite vertex. If the edges of the die have length 1, what is the shortest distance the ant needs to travel?

We unfold a cube to get a cross-shaped figure. Then, the problem is to find the shortest path between two points separated by a horizontal distance of 2 units and a vertical distance of 1 unit.

It is easy to see that the path in question is the one passing through the middle of the edge between the start and end points, and which has a distance of √2.

A Square and an Invisible Point

There is a square drawn on a piece of paper and also a point marked with invisible ink. You are allowed to draw 3 lines on the paper and for each of them you will be told whether the point is on its left, on its right, or lies on the line. Your task is to find out whether the point is inside the square, outside the square, or on its boundary. How do you do it?

Draw one of the diagonals of the square. Then, draw the 2 lines containing the sides of the square that are on the same side as the invisible point.

Five Points, Ten Distances

Five points, A, B, C, D, and E, lie on a line. The distances between them in ascending order are: 2, 5, 6, 8, 9, X, 15, 17, 20, and 22. What is X?

We assume that the points are ordered A to E from left to right. We have AE = 22 and either AD = 20, BE = 17, or AD = 17, BE = 20. Without loss of generality AD = 20, BE = 17, and therefore AB = 5, BD = 15, DE = 2. The distance of 6 is associated with either BC or CD, and therefore the points are arranged in one of these two ways:

  1. AB = 5, BC = 6, CD = 9, DE = 2
  2. AB = 5, BC = 9, CD = 6, DE = 2

If it is the latter, we get the sequence of distances: 2, 5, 6, 9, 11, 14, 15, 17, 20, 22, which does not fit the provided sequence.

If it is the former, we get the sequence of distances: 2, 5, 6, 8, 9, 14, 15, 17, 20, 22, and therefore X = 14.

Wacky Wordies 2

Can you figure out what phrases and sayings are represented in the Wacky Wordies below?

The answers are:

  1. Leave no stone unturned
  2. Foot in the door
  3. Go on a double-date
  4. Green with envy
  5. Look square in the eye
  6. Broken promise
  7. Pull up alongside the curb
  8. Excuse me
  9. High-grade performance
  10. Take on a big job
  11. Split the difference
  12. He came out of nowhere
  13. Wait on hand and foot
  14. Suit to a T
  15. Know it forward and back
  16. A period in history
  17. Crooked lawyer
  18. Get up on the wrong side of bed
  19. Sign on the dotted line
  20. Disorderly conduct