Puzzle of the Week #548 - Fifty Pebbles

You have a bag that contains fifty pebbles, twenty-five painted white and twenty-five painted black.

You randomly remove two pebbles.

If either of them is white, discard it and put the other pebble (whether it is black or white) back into the bag.

If both are black, discard both of them.

You repeat this process until there is only one pebble in the bag, what is the probability that the final pebble is white?

Puzzle of the Week #547 - Fourth Triangle

A Pythagorean triangle can never have a 22.5 degree angle, since tan(22.5) is 1-sqrt(2) which is irrational, however a 5,12,13 triangle comes very close. (It isn’t surpassed in accuracy until it is marginally improved in a triangle whose shortest side is 3648).

If three triangles with sides in the ratio 5,12,13 are placed with their smallest angles together as shown, what Pythagorean triangle has the necessary angle to complete the right angle?

Puzzle of the Week #544 - Exact Expression

a, b and c are each real numbers between 0 and 1.

 

I have a number ‘a’ and I want to find an exact expression for it.

I take the reciprocal and get that 1/a = 1+b.

I subtract the 1 and take the reciprocal. I find that 1/b = 3+c.

I subtract the 3 and take the reciprocal. I discover that 1/c = 4+b.

 

What is the exact value of a?

Puzzle of the Week #541 - Merging Lanes

A series of self-driving cars are driving along a road such that the distance between the cars is precisely the minimum recommended according to the formula:

 

Overall stopping distance (ft) = (Speed(mph)^2)/20 + Speed(mph)/2

 

So that at, for instance, at 40mph, the distance would be (40^2)/20 + 40/2 which is 100ft.

 

For simplicity let’s ignore the length of the cars themselves.

 

Two lanes that are each travelling at 60mph merge into a single lane. What speed will the cars in the merged lane be travelling at?

Puzzle of the Week #540 - Maximum and Minimum

There are 108 possible arrangements of the digits 1-8 such that the number formed by the first two digits is divisible by 3, the third & fourth digits divisible by 4, the fifth and sixth digits divisible by 5, and the seventh and eighth digits divisible by 6.

For example 42763518: 42 = 3x14; 76=4x19; 35=5x7; 18=6x3.

Now if I were to add together those multipliers 14, 19, 7 and 3 I would get a ‘score’ for 42763518 of 43. I did the same for all 80 possible arrangements and found the maximum possible score was 62 and the minimum was 28.

What were the arrangements that led to those scores?

 

Puzzle of the Week #538 - Sequence x Segment

When one solver (Roger) asked this week how I came up with the puzzle about the recurrence sequence my surprising answer was that it emerged from investigations into the semicircles-in-a-segment puzzle from the week before.

Here I unite the two.

Start with a sequence where after an initial fair of 1s each term is 4x the previous term minus the term before: 1,1,3,11,41,etc.

A second sequence is the product of pairs of consecutive terms in that sequence: 1,3,33,451,etc.

Finally form a series of semicircles whose radii are the reciprocals of the numbers in the product sequence, then place them in a segment of a circle such that the diameters of the semicircles lie on the chord of the segment.

What is the radius and central angle of the segment that exactly fits all of those semicircles?

Puzzle of the Week #537 - Non-Prime Sequence

A sequence starts 1,3,21 and then every subsequent term is 5 times the sum of the previous two terms, less the term before that, so that the fourth term is 5*(21+3)-1 = 119, and the fifth term is 5*(119+21)-3 = 697, etc. The sequence continues:

1,3,21,119,697,4059,23661,137903,803761,4684659…

None of those numbers after 3 is prime.

In fact I know for certain that 3 is the only prime number in this sequence and that each of the infinity of subsequent terms is composite.

How can I possibly know that?