# Bogglers

By Scott Kim|Thursday, June 01, 2000

Skydiving on Iltoz

 Illustration by Mark Matcho

The most popular sport on the imaginary planet Iltoz is skydiving. In the formation shown at right, every skydiver holds the hand or foot of two others. Notice that the only feet not held are on the outside. Iltoz skydiving guidelines, however, require that every foot be held by a hand and that every hand hold one foot. An easy way to do this is to have one person hold both feet of another to form a ring, as shown at left.

 Illustration by Mark Matcho

1) Advanced skydiving regulations require each sky diver to connect with four other sky divers. How many are required in the smallest formation?

2) The best solution to question #1 requires the sky divers to cross over themselves and twist their bodies in complicated ways. Barring such contortions, how many people are needed to make up the smallest formation, and what is the smallest number of sky divers that can be arranged into two completely different formations?

Constellations of Iltoz

 Illustration by Mark Matcho

The five letters of the Iltozian language each contain four dots, as shown below, making a total of 20 dots. Coincidentally, the stars of the night sky are grouped into constellations of 20 stars each, as shown in figures 1 through 4 below. For each constellation, divide the 20 stars into five groups of four to form the letters I L T O Z.

A sample constellation is shown at right. Letters may be rotated at any angle or uniformly enlarged but not otherwise distorted (the O in the sample has been enlarged and rotated slightly counterclockwise). As the T in the example demonstrates, the letters may overlap as long as each star belongs to only one constellation.

 Illustration by Mark Matcho

Satellites of Iltoz

 Illustration by Mark Matcho

Planet lltoz is a cube. The Iltozians want to launch enough satellites into orbit so that a person standing in the center of any one of Iltoz's six square faces is able to see exactly one satellite. The satellites will be in geostationary orbit: Each will hover in place above a single point. A satellite hovering above a corner would be seen by Iltozians on three faces of the planet. Two satellites placed on opposite corners would be seen by all on Iltoz, as seen in the illustration above.

1) How many satellites must be launched for a person in the center of any of Iltoz's six faces to see exactly two satellites? Those on different faces must not be able to see the same pair of satellites.

2) How many satellites are needed for Iltozians to see three from any face of the planet? Again, people standing on different faces must not see the same combination of satellites.

3) How many satellites are needed to see four?

Solution

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