289: Food Drive Cranberry Sauce

Quick, how many cans in the pyramid?

Same number of cans, different arrangement.

Because it's Christmas ...

215: Units Digit Redux

What's the best way to go about figuring this one out? Is it like the previous one?

Source:

.@MATHCOUNTS Math Challenge Problem #mathchat pic.twitter.com/071Kvg8smg
— MathMovesU (@mathmovesu) July 10, 2014

214: Last Digit

What is the last digit (units digit) in the expansion of 3⁹⁹⁹⁹?

How can we find this digit with tech?
Without tech?

Which way is more "clever"?
Which way gives you more satisfaction?

205: Special Triangle Spiral

If the side marked "x" is considered the beginning, and this pattern continued CCW, getting larger infinitely, how many sides would be of integer length?

187: Spiral

If we kept going, would any of the sides in this figure be parallel?

UPDATE: Asking a bad question is my fault. Of course, any polygon that has an even number of sides has n/2 pairs of parallel sides. What I was originally after, but didn't ask properly, was whether there a side of one polygon that was parallel to a side in another ... other than the pair in the triangle and square.

139: Largest Integer, part 2

After yesterday have to ask ... Is there a largest odd number that is not the sum of three composite odd numbers?
— James Tanton (@jamestanton) May 26, 2014

138: Largest Integer

What is the largest even integer that cannot be written as the sum of two odd composite numbers?

For example, 42 = 21 + 21, so it is not a candidate. 22 is a candidate because no pair of 9, 15, or 21 can equal 22.

As with many of Mr. Tanton's puzzles, there's a way to know that you are absolutely correct. Can you find the number and the explanation?

What is the largest even integer that cannot be written as the sum of two odd composite numbers?
— James Tanton (@jamestanton) May 25, 2014

137: McNuggets

McNuggets come in boxes of 3, 6, or 20. You can get 9 by buying a 3-piece and a 6-piece, but you can't get 13.  You can get 21 by buying seven 3-piece boxes, but not 22. You can get 23 with a 20-piece and a 3-piece, etc.

What is the largest number of McNuggets you CAN'T get with some combination of 3-piece, 6-piece or 20-piece boxes?

104: Recursive Pattern

Area at step 12?

93: Diagonals and Patterns

Step 43?

Each vertex of a n-sided polygon connects to how many other vertices? And count each diagonal only once ...

84: Choices, choices ... Lottery Winnings

You won the Lottery, but not the top prize. Instead, you get to choose which way you get paid your winnings (and we're going to assume "no taxes" because you're awesome). You will be paid for 3 weeks (7 days a week) plus one more day for good measure, so a total of 22 days. You get to choose from these options:

1. You get $100 for the first day, $200 for the second day, $300 for the third day. Each day you are paid $100 more than the day before.
2. You get 1 cent for the first day, 2 cents for the second day, 4 cents for the third day. Each day you are paid double what you were paid the day before.
3. You get 1 dollar for the first day, 1 dollar for the second day, 2 dollars for the third day. Each day you are paid the sum of the two previous days.

Which do you choose?

73: Four-Squares

2² ends in a 4
12² ends with 44
Find a square that ends with 444.
Find another that ends with 4444.

2^2 ends in a 4. 12^2 ends with 44. Find a square that ends with 444. One that ends with 4444?
— James Tanton (@jamestanton) March 13, 2014

62: Basketball.

There are 68 teams in the U.S. College Basketball Tournament.  The bottom eight play an extra round; the winners get the bottom seeds in the main tournament.  The main bracket has 64 teams in a single elimination tournament.

In the main bracket, how many games will be played, total?

If you just flipped a coin, what is the probability that you'd win them all?

Warren Buffet has offered $1 Billion Dollars to anyone who does pick all the games. What are the chances he will lose 1/60th of his money?

If you'd like to join the fun. with the rest of the Mathtwitterblogosphere.

26: The Units Digit

Let's play "Spot the Pattern" !

What is the units digit of $3^{3^{3^{3}}}$ ?