492: Sneak in Some Algebra

Marilyn Burns pointed out:

Well, does it?
Is there a pattern that always works?


source.

442: Square Root

x² = 81 has two solutions, -9 and 9.

But does √81 have one solution or two? Is it correct to say that √81 = +9 and -9?

Or should we be saying that √81is an expression and that 9, 18/2, 27^(2/3), and 1+6+2, are equivalent expressions?

It is really alarming how often I see materials written by math teachers claiming that for example sqrt(81) is both 9 and -9.
— Kate Nowak (@k8nowak) April 22, 2015

441: How many K?

Edited out the answer and some hashtags  ...

For how many integers k is 10,000 - k² positive?

For how many integers k is 10000-k^2 positive? #SATPrep #mathchat  @MathCurmudgeon
— David Marain (@dmarain) April 17, 2015

434: Integer Quotient

Is it possible to find a positive integer value of  n so that this quotient is an integer?

419: The Value of Why

Can you find values for y that will make each answer true?
Generalize the rules here.

418: Which is larger?

Which is true?

Left side is larger than Right side

Right Side si larger than Left Side.

The two sides are equal.

Not enough information?

How can we deal with this WITHOUT a calculator?

408: Balls

source.

386: Finding Primes the Euler Way

Trying to find a prime number seems difficult. You have to test it against divisibility by all of the primes that are smaller. But what if there were a formula?

Take a counting number, multiply it by itself, subtract it from the result and then add 41

1x1 = 1; 1-1 = 0; 0+41 = 41. That's prime.
2x2 = 4; 4-2 = 2; 2+41 = 43. That's prime.
3*3 = 9; 9-3 = 6; 6+41 = 47. That's prime, too.
4*4=16; 16-4=12; 12+41= 53. That's also prime.

How good is this idea?
Will it ever fail?

306: Factorials and Perfect Squares

Let's examine the function g(n):

g(n) = smallest integer such that g(n)*n! is a perfect square.

How should we go about finding if there's a pattern in that?

Plot of log(g(n)) versus n @jamestanton g(n) = smallest integer such that g(n)*n! is a perfect square. pic.twitter.com/tUvhiR0LjI
— Republic of Math (@republicofmath) October 18, 2014

289: Food Drive Cranberry Sauce

Quick, how many cans in the pyramid?

Same number of cans, different arrangement.

Because it's Christmas ...

288: Orange Pile

Quick ... more limes or more oranges?

282: Number puzzle

"The sum of two positive integers is 216. The greatest common factor of the two numbers is 24. What are all the possible pairs of numbers?"

What approach seems the easiest here?
I can see using solution methods such as:

• algebra
• guess and check
• organized list
• visual representation

Which (or which other) method rings true for you?
Which of the two sentences eliminates the most numbers?

The sum of 2 pos int is 216 and their gcf is 24. Find all possibilities.
— David Marain (@dmarain) September 24, 2014

276: Primes' Product and Sum

The really difficult questions on the SAT (the ones at the end of each section) are often not really that difficult -- they just seem to be hard to answer because they're hard to read and understand because of the formal mathematical writing style and the complexity of the sentence. Often the best tactic is to plug in some numbers for the vague-sounding variables and make the problem more concrete.

"The positive number k is the product of four different positive prime numbers.
If the sum of these four prime numbers is a prime number greater than 20,
what is the least possible value for k?"

So what can we do with this one to make it simpler to understand the question?
Where would you start?

247: SAT Number theory

This is an actual SAT question, from later in a section so it is one that a lot of test-takers got wrong because it was a hard question, was worded strangely, had too many undefined letters, or was near the end of a section and the test-takers simply ran out of time.

What are some of the difficulties you think most students would have with it? What are some things we could do to rewrite the question so it's more clear?

245: Multiplication

What are some different ways we might find the greater product here?

35 * 29  OR  39 * 25

Can you do this algebraically?
Graphically?

source: Fawn Nguyen @fawnpnguyen

243: Powers and Squares

The process seems to be the interesting thing here. How would you begin to work on this?

In how many ways can you write 2ⁿ as a difference of two squares?

source:

In how many ways can you write 2^n as a difference of two squares?
— James Tanton (@jamestanton)

231: Friedman Numbers

How do you suppose we could go about finding these?
Any methods or algorithms that we might use?

Source.

Challenge: How would we determine whether this is an infinite set or a finite one?

229: Create a special 10-digit number

Can you create a 10-digit number, where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number?

Some hints:
Can it be 0000000000?
Can it be 9000000000?

Source: Richard Wiseman.



One answer, slightly obfuscated, is 2³ * 5³ * 7 * 887143

210: Rationally Radical

We all know it's possible to multiply two radicals together and get an integer result.
( √7 ) * ( √7 ) = 7 is one possibility.

Is it possible to ADD two or more radicals and get a rational result?

184: Is Eight Magic?

What is the pattern if we do this for other two-digit numbers?

Can we get other results? Are there results that we CAN'T get?
Challenge:
   What is the longest string?