BrainDen.com - Brain Teasers

# bonanova

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2. ## Two boys

I ask people at random if they have two children and also if one is a boy born on a tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.
3. ## Shortest set of lines in a square, revisited

Recently we considered the shortest roadway that connects the four corners of a square. Here we seek the shortest set of line segments, one attached to each of a square's corners, that need not connect with each other. Instead, what we ask of the line segments is that is they will block any ray of light attempting to pass through the square.
4. ## a^b x c^a = abca

Find a, b, c. ab x ca= abca , a 4-digit number

Clue.
6. ## Two boys

@ThunderCloud you're homing in on it, but now you're a little high. @BMAD It's certainly true that if the FIRST child was a boy born on a tuesday, then it's just the prob that the second child is a boy. But ... the OP does not tell you that. That is, "one is a boy" does not imply "my oldest child is a boy." So your "second" child simply means the "other" child.
7. ## a^b x c^a = abca

Not that I know of. Tables of powers, or spreadsheet where different abc values can be simply typed in, or inspired guesswork? It's not my fav type of puzzle, but some like this type.
8. ## Two boys

@ThunderCloudThat's close, but a bit low.
9. ## The Most Wonderful Prince

So this guy Thomas Bruss solved the general stopping problem.
10. ## The Most Wonderful Prince

The King has decreed that his daughter the Princess shall marry the most wonderful Prince in all the land. One hundred suitors have been selected from their written applications, and on a certain day the King arranges for them, in turn, to interview the Princess. Each suitor must either be chosen or eliminated on the spot. If the Princess does not choose any of them, she will marry the last Prince to speak with her. You have been chosen as the Royal Advisor to the Princess and tasked with implementing her best strategy to choose the Most Wonderful Prince of the realm. You devise an evaluation scheme by which the princess can assign a unique "wonder number" to each prince as she meets him. The strategy then is to have the Princess reject, but record the highest score of, the first N princes that she meets. The Princess will then choose the first Prince that she subsequently interviews whose score exceeds that recorded score. That's it. The puzzle is basically solved. Except, of course, to decide on the optimal value of N. It requires some thought. If N to too high, the most wonderful prince is likely to be eliminated at the outset, and she ends up with the last guy. If N is too small, the Princess will likely settle for a fairly undistinguished prince. What value of N optimally balances these two risks? What is the probability that the Most Wonderful Prince will be chosen? Disclaimer: I recall this puzzle being posted before, with different flavor text. And it's somewhat of a classic. To give it a fair play here, I'll ask not to post any links and not to just give the answer if you know it, at least not without "showing your work."

12. ## Salesman in a Square

Al made sales calls at a number of cabins, which lay in a square field, one mile on a side. He drove his car to the first cabin, then visited the remaining cabins and returned to his car on foot, walking several miles in the process. If Al had had a map and perhaps a computer, he could have picked the shortest route to take, (traveling salesman problem). Lacking these amenities, Al simply chose to visit (after the first one) the (un-visited) cabin closest to his current location. If there were 6 cabins in all, how might they be placed, so that using Al's nearest-neighbor algorithm, and selecting the worst initial cabin, Al would be forced to walk the greatest distance; and what is that distance? Examples: 2 cabins: diagonal corners, starting at either: 2 x sqrt(2) = 2.828... miles. 3 cabins: any three corners, starting at any of them: 2 + sqrt(2) = 3.414... miles. Check out n=4 and n=5 as a warm-up.
13. ## The Most Wonderful Prince

@plasmid Not sure why derivative failed, but you're right about the result. Nice solve.

15. ## Random passengers

These cases are the ones I calculated as well, and led me to the right conclusion. (I had to solve it as it's not mine originally and I did not receive the solution.) It took a little insight to get me thinking along productive lines. See the spoilers in my April 1 post - btw not an April Fools post - to get there.

17. ## The Most Wonderful Prince

Great job! Want to look at the reciprocal of that number and guess the exact result?
18. ## Fettered random walk

@Molly Mae blazed the trail and @plainglazed nailed it.
19. ## Fettered random walk

Consider a random walk in the plane where each step is taken, beginning at the origin, in either in the positive x or positive y direction, i.e. either east or north, each choice being made by the flip of a fair coin. The length of each step is 1/2 the length of the previous step, and the first step has length √2. After infinitely many steps have been taken, what is your expected distance from the origin? Edit: Ignore the original text in pink. Instead, What is the distance to the origin of the centroid of the possible termination points? You find the centroid of a set of points by averaging respectively their x- and y- coordinates. First correct answer wins, but style points will be awarded as well.
20. ## Five rings

You are given 5 circles, A, B, C, D, and E, whose radii are, respectively, 5", 4", 2", 2", and 1". Can you find a way to overlap circle A with portions of some or all of the other four circles so that the un-overlapped portion of A has the same area as the sum of the unoverlapped portions of the other four circles? That is, the red area is equal to the sum of the green areas. Circles B, C, D and E may overlap portions of each other as well as a portion of A.
21. ## Fettered random walk

You are both on the right track, but I realize now that I mis-stated the OP. I didn't ask for what I wanted. What I wanted to get at was the average location, that is the average of all the possible ending location coordinates, more precisely, their centroid, and its distance from the origin. That's not the same as the expected distance of the ending points -- which does take sort-of serious math. My bad. I edited the OP.

23. ## I might be a memory aid

Sir, I bear a rhyme excelling In mystic force and magic spelling Celestial sprites elucidate All my own striving can't relate. See, I have a rhyme assisting My feeble brain, its task ofttimes resisting.
24. ## Who finished 2nd?

Al, Bert, and Charlie competed in a track and field event in which points were awarded for 1st, 2nd, and 3rd, place only. At the end of the day, Al had accumulated 22 points, while Bert and Charlie each garnered only 9 points. No other competitor earned points. Bert was 1st in the shot put. Who finished 2nd in the javelin throw? This is a Gold star puzzle.

Yes and yes.
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