Tag Archives: triangle

Triangle Sum Puzzle

This is probably a consequence of being a mathematician, but I have always enjoyed number puzzles. I think that there is a general simplicity and universality in numbers that are not present in things like word puzzles, where the ability to reach a solution can be limited to the vocabulary of the user.

The fact that these are puzzles and not simply homework exercises also helps because we often find people sharing difficulties and successes stories over the water cooler or at the lunch table. The fact that many of these math puzzles can teach some of the same concepts as homework problems (in a more fun and inclusive way) is generally lost on the user as their primary interest is generally on solving the puzzle in front of them, or sometimes solving the more general form of the puzzle.

Today’s post is about a puzzle that was originally shared with me over a lunch table by a friend who thought it was an interesting problem and asked what I thought about it. I didn’t give the puzzle much further thought (he had correctly solved the puzzle) until I saw it again in “Algorithmic Puzzles” by Anany Levitin and Maria Levitin. It was then that I thought about the more general form of the puzzle, derived a solution for the problem, and decided to code it up as a script for my site.

Below is a link to the puzzle:

We have a set of random numbers arranged in a triangle and the question is to find the path of maximum sum from the root node (the top node) to the base (one of the nodes on the bottom row) with the rules that
(1) Exactly one number must be selected from each row
(2) A number can only be selected from a row if (a) it is the root node or (b) one of the two nodes above it has been selected.

For the sample
So for the sample problem in the picture, the maximal path would go through nodes 57, 99, 34, 95, and 27.

For more of these puzzles check out the script I write here and be sure to let me know what you think.

Triangle Trigonometry

Triangle Script Image

I haven’t forgotten about my pledge to focus more content here towards some of the areas I’ve been asked to tutor on recently. This latest one is designed to help users understand the properties of triangles. It is based on two laws that we learn in trigonometry: the law of sines and the law of cosines. Assume that we have a triangle with sides of lengths a, b, and c and respective angles A, B and C (where the angle A does not touch the side a, the angle B does not touch the side b, and the angle C does not touch the side c). These laws are as stated as follows:

Law of Sines

a
sin(A)
=
b
sin(B)
=
c
sin(C)

Law of Cosines
c2 = a2 + b2 – 2*a*b*cos(C)

We can use these laws to determine the sides of a triangle given almost any combination of sides and angles of that triangle (the only one we cannot determine properties from is if we are given all three angles, as this leads to many solutions).

The script generates random triangles, with different combinations of sides and angles revealed and the user’s job is to try to determine the missing sides. There is a button to reveal the solution, or if you’d like to see how we arrive at these values, you can check the “Show work” box.

Hope you enjoy.

Other Blogs covering this topic:
Mathematical!
Algebra 2 Trig