This document contains a collection of mathematical problems intended to stimulate curiosity. The problems cover a variety of topics including geometry, algebra, probability, and graphing. They range in complexity from simple to challenging. The final section provides advice for teachers on staying focused on the curriculum, creating fair tests, choosing grading scales, professional development, and finding fulfillment in their work.

1. Problems That Stir Mathematical Curiosity
A Collection
by
R. W. Alexander, Ph.D.
2004

2. Try This E. Nichols NCTM Demo
Roy & Bob were fishing at
the river by the RR
bridge. After no luck,
Roy decided to cross over
the bridge.
Just as Roy got 3/8 on the bridge, Bob hollered
“Train!! Run to either end , you’ll just make it.” If
Roy can run 15 mph, a) from where( A or B ) is
the train coming and b) how fast is the train going?
A B

3. Try This Fun Problem
A B
C
a
b
h
c
x y
Triangle ABC is a primitive Pythagorean triangle with
altitude, h, drawn upon the hypotenuse, c. Devise a
scheme and find a set of 5 integers such that (a, b, h, x, y)
may be described as a Pythagorean quintuple. If one set is
found, create a solution for all such quintuples.
Note: (a,b,c) = (3,4,5) is a primitive Pythagorean triangle,
where as (a,b,c) = (6,8,10) is not. Also, c = x + y.

4. Try This Min/Max Problem
A
B C
. .S
M
N
Points R and S lie in the
interior of triangle ABC.
P
A line is drawn from R to S, with the condition that
the line must touch each side of the triangle before
reaching its destination. Find a path so that d =
( RM + MN + NP + PS) is the minimum distance
between R and S.
R

5. Try The Elusive Angle Problem
A
B C
D
E
F
Triangle ABC is
isosceles, with angle
A = 20. Angle BCD =
50; Angle CBE = 60.
Using basic geometry,
not trigonometry, find
angle DEB.

6. Try This Non-routine Problem
A
B
C
D
Triangle ABC is isosceles with angle A = 100; If the
segment AB is extended so that AD = BC, find the
measure of angle D, without using trigonometry.

7. Try This Network Problem
In a plane, when 4 utilities are connected
to 4 houses, what is the minimum
number of crossings; ie, k(4,4) = ?= 4 ??
Verify that k(5,5) = 16 & k(5,7) = 36.
In general, derive a solution for the problem
when p utilities are connected to q houses; ie,
k(p,q) = ????

8. Model this probability problem
A concerned virgin went to see the wise
Gypsy about the possibility of marriage.
The Gypsy instructed the virgin to gather 8
daisies in one hand and clip the flowers.
Holding the 8 stems, strings were to be used
to tie pairs of stems together above and
below the hand. If she released the stems
and they made one large loop, she would be
married within the year. What is the
probability that the virgin will marry within
the year? Start with a simpler problem, say
2 or 4 daisies.

9. Using all of your algebra skills,
solve this problem.
Put “Trust” into your algebra skills and
solve this interesting problem:
x ^ 4 = 4 ^ x
Try a simpler problem and verify your
results with a graphing calculator.

10. Final Comments
Teachers, stay firmly within the curriculum.
Give challenging but non-threatening tests.
Choose a proper grading scale for each class.
Try The Great Grading Scale, it is close to perfect.
Grow professionally and expand your network.
Be patient with mundane minutia--it will pass.
Love your students and build “Trust” with TGGS.
Celebrate your successes, rewards, and retirement.
Thank God for the life you have to share with others
and the inspiration that others have shared with you.