Puzzle of the Week #557 - Palindromic Numbers

I learned the interesting fact the other day that any natural number, however large, can be written as the sum of at most three palindromic numbers. For instance, 7369 = 7227+131+11.

 Can you write the ten digit number 7,275,640,031 as the sum of three palindromic numbers?

 Single digit numbers count as palindromes (eg, 7) but numbers with leading zeroes do not (eg 0330).

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?