bonanova

Yes.

The OP does not constrain the boy's arrival to be on the minute. Between 7:58 and 7:59, for example, p decreases linearly from 1 to 0.8; it does not drop abruptly to 0.8 at 7:58. Similarly for the next two minute-intervals. The boy does not arrive later than 8:01. Make a graph of p vs boy's arrival time and find its average value.

I assumed there is, , but I'm not sure right now why that is. The power set (the set of all subsets) of a set of n objects has size 2n. For 9 digits, n=9. Then subtract off the empty set.

Do you want each bowl to end up with 1/3 liter?

Good job all. Several great answers.

I've given up on this one also.

Sure. Rectangular Parallelepiped

Is there a solution w/o equations? I mean more simple? You did away with equations nicely. Your last two sentences are the solution.

Hi Joe, and welcome to the Den. Pairing is the right first step. Can you make the pairs even simpler?

Consider an isosceles triangle with base 10 feet and sides 13 feet. Now imagine building a snowman inside that triangle. A snowman made of circles, not spheres. It's easy to sketch. The bottom circle is tangent to all three sides. The next circle rests on the first and is tangent to the two sides. Likewise the third, fourth, and so on. Actually, there are an infinite number of circles in our snowman. But there must be a geometric series involved, or something like one, because the snowman never gets outside the triangle. So, it's fair to ask: What is the sum of the circumferences of all the circles? This is actually an unadvertised "aha" puzzle, so can you give the answer without writing anything down except for the sketch?

What is the area of the smallest square that holds five unit squares without overlap?

There's an even simpler observation that immediately gives the correct answer.

Evaluate 1002 - 992 + 982 - 972 + 962 - 952 + ... + 22 - 12 = ?

Write down all the mathematical symbols you know, but exclude any (e.g. cube root) that contain a number. Use as many as you like in as many ways as you like on the arrays of numbers below to make a valid equation. One of them is filled in for you to give an idea for solving the others. All but 3 - you belong at the Den All but 1 - you're a free-thinking Denizen All 10 - Take a bow and a star. 0 0 0 = 6 1 1 1 = 6 2 + 2 + 2 = 6 3 3 3 = 6 4 4 4 = 6 5 5 5 = 6 6 6 6 = 6 7 7 7 = 6 8 8 8 = 6 9 9 9 = 6

Correct. The latter: positive integers whose digits are strictly increasing left to right.

What do 15 489 1256 and 24578 have in common? They are positive integers whose digits are strictly increasing left to right. How many of them are there?

I'm not sure how to answer this, but consider that the average of the first 2n integers is (2n+1)/2, not n. It is larger by 1/2. Yes. You'd need to include zero to make it n. My bad.

I'm late marking this one solved. Nice, clear approach.

Nice thinking. Away with equations!

OK someone help me out on this. If the cases are 1 ... N/2, and there are TWO cases for all of them but N/2, for which there is only ONE case, then why isn't the average slightly LESS than N/4? And since this is my 4999th post, does that mean the next one has to be profound in some way? I'm in trouble. :~}

Nice. I actually stayed up last night and got it. I got the 7s in rows 234 ok. I attacked row 1 by trying stuff for cells 7 and 8. Once I made the right guess and CORRECTLY looking at the consequences, the 1289 cells opened up as you said.

A boy walks out on a train trestle high above a river. He has walked 4/7 of the way across the trestle when he spots a train coming at him head-on. He realizes that he has just enough time either to run toward the train and jump off the trestle or to run away from the train and get off the trestle. If the boy can run 6 miles per hour, how fast is the train moving toward him? Can you answer without writing anything down?