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Geometry and Logic Challenges

The document contains 10 geometry and logic puzzles involving arrangements of objects like toothpicks, pennies, and knights on a chessboard. Problem 1 asks how to determine which of two doors leads to freedom using one question to a sentry that always lies or tells the truth. Problem 2 asks how to arrange toothpicks to form four squares of equal size by moving two toothpicks. Problem 3 asks if six pencils can be arranged to touch each other and how it can be done.

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Tushar Walia
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414 views4 pages

Geometry and Logic Challenges

The document contains 10 geometry and logic puzzles involving arrangements of objects like toothpicks, pennies, and knights on a chessboard. Problem 1 asks how to determine which of two doors leads to freedom using one question to a sentry that always lies or tells the truth. Problem 2 asks how to arrange toothpicks to form four squares of equal size by moving two toothpicks. Problem 3 asks if six pencils can be arranged to touch each other and how it can be done.

Uploaded by

Tushar Walia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Geometry Puzzles

Problem 1 (But first, one last logic puzzle). Suppose that you are a prisoner, and you are con-
fronted with two doors: one leading to freedom, and one leading to the executioner’s chamber, but
you don’t know which is which. A sentry guards each door. You know that one sentry always lies,
and one sentry always tells the truth. Again, you don’t know which is which. You are allowed to
ask one of the sentries one question. What is a question you ask a sentry that will tell you with
certainty which door leads to freedom?

Problem 2. Toothpicks are arranged as shown below. Move two toothpicks to form four squares
each with side length equal to one toothpick.

Figure 1: Toothpicks, move two.

Problem 3. Is it possible to arrange six pencils so that each pencil touches each of the others? If
so, how?

1
Problem 4. Toothpicks are arranged as shown below. Remove four toothpicks to leave two equi-
lateral triangles. Remove three toothpicks to leave, again, two equilateral triangles. Finally, remove
just two toothpicks to leave two equilateral triangles. Loose ends are not allowed.

Figure 2: Toothpicks, remove four, three or two, leaving two equilateral triangles.

Problem 5. Ten pennies are arranged as shown below. What is the minimum number of pennies
we must remove so that no three of the remaining pennies lie on the vertices of an equilateral
triangle.

Figure 3: Pennies.

Problem 6. Draw six lines segments to form eight triangles. See if you can find two different ways
of accomplishing this.

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Problem 7. A river that is 4.0 meters wide makes a 90◦ turn as shown below. Is it possible to
cross the river by bridging it with only two planks laid flat, each 3.9 meters long. If so, how?

River

Figure 4: River and Planks.

Problem 8. In the distant future, regular travel between planets will become possible. Suppose
spacecraft travel along the following routes: Earth–Mercury, Pluto–Venus, Earth–Pluto, Pluto–
Mercury, Mercury–Venus, Uranus–Neptune, Neptune–Saturn, Saturn–Jupiter, Jupiter–Mars, and
Mars–Uranus. Can a traveler get from Earth to Mars?

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Problem 9 (Think about using graphs). A chessboard has the form of a cross, created from a 4 × 4
chessboard by deleting the corner squares. Can a knight travel around this chessboard, using the
usual knight’s move, passing through each square exactly once, and end up on the same square it
starts on?

Figure 5: Knight’s tour.

Problem 10 (Again, think about using graphs). Four knights are positioned on a 3 × 3 chessboard
as shown on the first chessboard below. Can they move to the positions shown on the second
chessboard?

I II

Figure 6: Knight’s positions, before and after.

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