Today’s Puzzle:

Why do I want a hip hypotenuse for Christmaths?

Many have heard the equation a² + b² = c² to help find the hypotenuse of a right triangle when given two legs. What do you do if you are given the hypotenuse and a leg instead of two legs? You use b² = c² – a².

Sometimes finding squares and taking square roots is not too difficult:

• x = √(29²-21²)                              x = √(61² – 11²)
• x = √(841-441)                           x = √(3721 – 121)
• x = √400                                         x = √3600
• x = 20                                              x = 60

Other times it can be more challenging:

• x = √(177² – 48²)
• x = √(31329 – 2304)
• x = √29025
• x ≈ 170.37, that’s irrational and not in the simplest form. Finding the factors of 29025, so its square root can be simplified, is going to be a pain!

Try this instead:

• x = √(177² – 48²) That’s the difference of two squares, so it can be factored!
• x = √((177 – 48)(177 + 48))
• x = √(129 · 225) I love that I have two factors instead of one big number! And in this case, one of them is a perfect square! 225 = 15².
• x = √(3 · 43 · 15²)
• x = 15√129.

Most people learn the Pythagorean theorem before they learn how to factor the difference of two squares, but then they never apply it to the Pythagorean theorem. Once you know both concepts, factor whenever you can!

A Math Parody of the Song, “I Want a Hippopotamus for Christmas”

On Thanksgiving, my son gave me an early Christmas present: a t-shirt that had a right triangle with a hippopotamus sprawled over the hypotenuse. The shirt had the words, “I want a hippopotenus for Christmath” at the bottom.

Somebody suggested that I sing it.

I looked for a mathematical version of I Want a Hippopotamus for Christmas and couldn’t find one, so I made my own. I decided I liked “hip hypotenuse” better than “hippopotenuse” and that the British “maths” sounded better than the American “math.” Then I recorded it. If I had more time, I would have waited until I didn’t have a cold and would have worked on my timing a bit more. Since I wanted it to be ready to present on a certain day at school, I just went with it as is. I hope you enjoy it.

I shared it with my family. Here’s how one of my sons responded:

Well, some moms have made negative comments about their kid’s ability to learn some math concepts. Indeed, I didn’t.

I already had been thinking about reworking some of the lyrics, making the first half of the song about the Pythagorean theorem and the second half about Trigonometry. I’ll re-record it sometime, but here are the lyrics I’ll use for the revised version:

I want a hip hypotenuse for Christmaths.
Only a hip hypotenuse will do.
I don’t want a doll, no dinky tinker toy;
I want a hip hypotenuse to play with and enjoy!

I want a hip hypotenuse for Christmaths.
I don’t think Santa Claus will mind, do you?
He won’t have to use those squares to find the last side, too.
Just the difference times the sum. That’s an easy root to do.

I can see me now on Christmaths morning calculating squares.
Oh, what joy and what surprise
When I open up my eyes
To see that hip hypotenuse given there.

I want a hip hypotenuse for Christmaths,
Only a hip hypotenuse will do.
No crocodiles, eating more or lesses.
I only like hip hypotenuses.
And hip hypotenuses like me, too.

I want a hip hypotenuse for Christmaths.
A hip hypotenuse is all I want.
Mom says triangles are often right, for them
Teacher taught a theorem that is Pythagorean.

I want a hip hypotenuse for Christmaths,
The kind that’s used in trigonometry.
The sine of an angle is its opposite side
Over the hip hypotenuse. Make sure it’s simplified!

I can see me now on Christmaths solving triangles downstairs.
With the laws of sines and cosines.
Each time(?!) I must choose
When there’s no hypotenuse anywhere.

I WANT a hip hypotenuse for Christmaths,
Only a hip hypotenuse will do.
No crocodiles, eating more or lesses.
I only like hip hypotenuses.
And hip hypotenuses like me, too.

I hope my song made you laugh. For more laughs, check out this Statistics Saturday post at Another Blog, Meanwhile. He lists several humouous hippopotamus-related unwise Christmas gifts.

Factors of 1769:

This is my 1769th post. What are the factors of 1769?

• 1769 is a composite number.
• Prime factorization: 1769 = 29 × 61.
• 1769 has no exponents greater than 1 in its prime factorization, so √1769 cannot be simplified.
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1769 has exactly 4 factors.
• The factors of 1769 are outlined with their factor pair partners in the graphic below.

More About the Number 1769:

1769 is the sum of two squares in two different ways:

40² + 13² = 1769, and
37² + 20² = 1769.

1769 is the hypotenuse of FOUR Pythagorean triples:
319-1740-1769, which is (11-60-61) times 29,
969-1480-1769, calculated from 37² – 20², 2(37)(20), 37² + 20²,
1040-1431-1769, calculated from 2(40)(13), 40² – 13², 40² + 13²,
1220-1281-1769, which is (20-21-29) times 61.

Did you notice that 20-21-29 and 11-60-61 were the two triangles used in today’s puzzle? It would not have been so easy if I had used 319-1740-1769 and 1220-1281-1769 instead!

1769 is a palindrome in some other bases:
It’s 585 in base 18,
1I1 in base 34, and
TT in base 60.