Daily Brain Puzzlers Calendar

Darts

Consider a dart board made up of four little circles of radius 1 inside a large circle of radius 2. The four little circles meet at the origin, which is the center of the large circle. The centers of the little circles are on the four axes. The area in question is outside of all the little circles, but inside the large circle.

George has trained his pet monkey to throw darts at the area in question. Although the monkey always hits the board, his dart just lands at a random point on the board. What is the exact probability that his dart will land in the area in question? Assume that the probability that a dart will land in a given area is proportional to the ratio of that area to the total area of the board.

based on a Puzzle Corner by Allen Gottlieb in Technology Review, via The Bent

Well Iā€™ve got it all laid out. I just think I donā€™t know one important equation.

Summary

I believe the answer is Overlap/Pi (where overlap is one instance of the overlapping area of two of the small circles. My estimate of that area is about 0.5. So call it 16%?

= (.5-1/ :pie:) is roughly 18.169%

1 Like

Correct!

Spoiler hint if you like it

The overlap is kind of a red herring. You can split the 4-circle area into a square and 4 semicircles and not have to consider the overlap. Itā€™s not so obvious if you donā€™t draw to scale or or label points.

For the record, my solution dealt directly with the overlap, as I found it not too hard to deal with.

I love solving different puzzles, and itā€™s always fun to challenge ourselves. Even if you canā€™t post daily, Iā€™m sure itā€™ll be exciting to see the puzzles and try to solve them together.By the way, have you ever tried online jigsaw puzzles? Theyā€™re a great way to relax and exercise your brain. I recently came across a website that offers a wide range of jigsaw puzzles. Itā€™s a fantastic way to unwind and have some fun!Iā€™m new to this forum, and itā€™s awesome to connect with fellow puzzle enthusiasts. Looking forward to joining in on the brain-teasing fun.

Ladder Locations

In the game of Chutes and Ladders, if you land on certain square you can advance or retreat many squares. In this problem, we are considering a variation in which there are only ladders.

The board is a 10-by-10 grid with the bottom row numbered 1 to 10 from left to right, the second row numbered 11 to 20 from right to left, the next row numbered 21 to 30 from left to right, etc. ending with 100 in the top left corner. Each ladder is a straight line from the center of one square up to the center of another square. No ladder is vertical and no two ladders touch or cross each. If a ladder, for example, started in the second row and ended in the seventh row, it would jump over four rows.

In this variation, there are seven ladders, each of which jumps at least one row. In fact they each jump a different number of rows. Each ladder starts in a square whose number is a perfect square greater than 1, and ends in a square whose number is a prime. Give the numbers of the starting and ending squares of the seven ladders, in increasing order of the starting squares.

From An Enigma by Susan Denham in New Scientist by way of The Bent

At first I missed the fact that the rows snakes left to right, right to left, etc. Without that, I donā€™t think you can do it. My solution below also relies on ā€œa straight line from the center of the squareā€ and therefore a very loose definition of ā€œnot touchingā€.

Summary

4-79
9-53
16-97
25-89
36-73
49-71
64-83

1 Like

Correct!

The M. G. G. C. (Mathematiciansā€™ Goofy Golf Club), to which I belong, always holds its annual
banquet in July, and I was trying to find the date from some fellow members who were being annoyingly obtuse. One told me the date was an odd number, another that it was greater than 13, a third that it was not a perfect square, and a fourth that it was a perfect cube. Finally, W. W. Webb, president of the club, told me that the date was less than my highest single hole score last season (which was, in fact, 17, and I thought it was a little tacky of Webb to bring that up). Later, when I finally learned the date, I discovered that only one of the five statements was correct. What was the date of the banquet?

From Brain Puzzlerā€™s Delight by E. R. Emmet by way of The Bent

MGGC

Summary

Itā€™s on the 4th. Since either >13 or <17 must be true, all the others must be false. So it must be an even perfect square. Either 4 or 16. And 16 would make both >13 and <17 true.

1 Like

You are correct, but your explanation is off.

Explain

I take it back. I was a little caught up on the ā€œnotā€ for falses. Any number from 1 to 31 is either over 13 or less than 17 or both. Both cannot be true. Therefore, one of those IS true, which means the other 3 clues must be false. I was thinking the number was either <13 or >17, which is true, so I had thought your inequalities were backwards, but I rescind my criticism. And I wonā€™t be a jerk and just delete my prior post. Iā€™ll stand up and say I was overly judgemental. Sorry.

Correct!

A candle in the shape of a truncated right circular cone 45 cm high burns to extinction in nine hours. The bottom 3 cm take 20 minutes longer to burn than the top 3 cm. How many minutes will it take the top 3 cm to burn? Assume that the volume of the candle decreases at a constant rate as it burns.

Marcello J. Carrabes via The Bent

Volume of a cone is (1/3) * pi * h * r^2

Volume of a truncated cone would be the difference of two cones

Radius varies linearly with height

Oh, btw, TIL: frustum is the portion of a cone or pyramid which remains after its upper part has been cut off by a plane parallel to its base, or which is intercepted between two such planes.

One equation for the volume of the frustum of a cone is (1/3) * pi * h * (r1^2 + r1*r2 + r2^2)

IS the answer 26? I feel like the answer should be 26.

Not exactly 26, but it rounds to that.