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.

Perfect Square Sums

Find all possible arrangements of the numbers 1 to 15 in a sequence, where the sum of any two consecutive numbers is a perfect square.

Note that the number 8 can be neighboring only with the number 1, so it must be at one end of the sequence. The number 15 can be neighboring only with the numbers 1 and 10, so it either needs to be next to 1 or at the other end of the sequence. In either case, 10 should be next to it. The only other number that can be neighboring 10 is 6. Then 3 should follow, then 13 (since 1 is already taken), then 12, then 4, then 5, then 11, then 14, then 2, then 7, then 9. Since 1+9=10 is not a perfect square, we find that the only solutions are


and its reverse.

Sunome Puzzles

The main challenge of a Sunome puzzle is drawing a maze. Numbers surrounding the outside of the maze border give an indication of how the maze is to be constructed. To solve the puzzle you must draw all the walls where they belong and then draw a path from the Start square to the End square.

The walls of the maze are to be drawn on the dotted lines inside the border. A single wall exists either between 2 nodes or a node and the border. The numbers on the Top and Left side of the border tell you how many walls are on that line of the grid. The numbers on the Right and Bottom of the border tell you how many walls exist in those rows and columns, respectively. In addition, the following must be true:

  • Each puzzle has a unique solution.
  • There is only 1 maze path to the End square.
  • Every Node must have a wall touching it.
  • Walls must trace back to a border.
  • If the Start and End squares are adjacent to each other a wall must separate them.
  • Start squares may be open on all sides, while End squares must be closed on 3 sides.
  • You cannot completely close off any region of the grid.

Examine the first example, then solve the other three puzzles.

The solutions are shown below.