Puzzle of the Week #587 - Greatest Quadrilateral

Last time, three friends each ran 25, 33 and 39 miles from the same staring point to maximise the area of the triangle formed by their endpoints.

This time, those same friends want to enlist a fourth friend. The three original runners will run the same distances as before. If they want the area of the resulting quadrilateral to be one and a half times that of the triangle from last time, what is the minimum distance must the fourth runner cover?

 

NB: if you didn’t solve the previous puzzle or don’t want to, please see the solution to that puzzle.

Puzzle of the Week #585 - Territorial Expansion

If I start with an equilateral triangle 10 miles on each side (area ~43 square miles), then move one of the corners 10 miles in order to maximize the new area, it’s obvious what I should do: move one of the corners 10 miles away from the centre of the original triangle. This would make my new territory an isosceles triangle with angles 75, 75, 30 (area ~93 square miles).

If instead I can move all three of the corners by 10 miles, I would move each of them 10 miles directly away from the triangle centre and get a larger equilateral triangle (area ~323 square miles).

If however, I could only move two corners, each by 10 miles, moving them directly away from the original centre of the circle, (giving an area of ~187 square miles and angles of 45,45,90), would not in fact give the maximum area.

It is possible to achieve a maximum area of over 191 square miles (moving only two corners 10 miles each), but what would the angles of the resulting triangle be in this case?

Puzzle of the Week #582 - Unique Number

There exists a number, uniquely as far as I can tell, that has the following property:

Its digits are summed in pairs in all possible combinations.

These pairwise sums are then all multiplied together.

The resulting product is the original number.

Can you find this special number?

 

For example, if the number was 125, the pairwise sums would be 1+2=3, 1+5=6, 2+5=7. The product would then be 3x6x7=126. But since this isn’t equal to 125, this is merely a near miss and not the number we are looking for.

 

Puzzle of the Week #581 - Inevitable Divisibilty

If you take any three whole numbers A, B and C and calculate the differences A-B, A-C and B-C, and then multiply those three differences together you can prove that the result will always be even. This is because of the numbers A, B and C, either at least two of them are even, or at least two of them are odd, and so at least one of the differences must be even.

If instead you consider a set of any 6 whole numbers A-F and do the same thing, what is the highest number that the resulting product will definitely be divisible by?

Just to be clear, the difference between every pair of numbers is considered, but only one way, so (A-B) would be included but not also (B-A).

Puzzle of the Week # 579 - Trip Counter

True story: when I bought my Dacia Sandero a few months ago I immediately reset the trip counter to zero. The overall mileage of the car was 55664. I noticed that I was only 0.1 miles into ownership when the overall mileage changed to 55665, so I can surmise it was actually on 55664.9 when I bought it, even though it only displays full miles. The trip counter displays the mileage to an accuracy of 0.1 miles, and if you don’t manually reset it can count as far as 9999.9.

At some point, will the overall mileage and the trip counter mileage appear the same, with the only difference the decimal point in the trip counter mileage? If so, what will that mileage be?

Puzzle of the Week #578 - Prime Bridge

The first nine prime numbers (2,3,5,7,11,13,17,19,23) are trying to get across a bridge. They only have one car, which they are each capable of driving, and which can hold a maximum of three of them. The time the car takes to cross the bridge is whatever the highest prime number in the car for that crossing, for instance if the 3, 7 and 17 are in the car it would take 17 minutes to cross.

What is the shortest time in which all the prime numbers can cross the bridge?

Puzzle of the Week #576 - Overlapping Chessboards

I take a pair of 10x10 chessboards. Initially I overlay them exactly on top of one another and find the point 4 squares along and 2 squares down and drive a nail through both boards. The I rotate the top board around the nail until the bottom left corner of the upper board coincides with the bottom edge of the lower board.

I take a second pair of 10x10 chessboards and do exactly the same except that this time the nail is driven through at a point 3 squares along and 1 square down.

The question is: in which scenario is the area of overlap between the upper and lower chessboards greatest?