Puzzle of the Week #458 - Phone Charger

This is my portable smartphone USB-C charger. It very handily and neatly docks onto the charging port of my android phone, without any need for a wire between them. To charge the charger there is a socket on the end as shown, which is also USB-C.

My question is, if you had a supply of chargers identical to this one, how many would you need to form a ring, where each charger was charging the next one in the chain?

Puzzle of the Week #457 - Sloping Square

Within a rectangle of area 882, a square of area 196 is drawn, aligned with the corner of the rectangle, and tangent to the diagonal of the rectangle as shown.

A second square is drawn, aligned to the diagonal, and tangent to the outside of the rectangle as shown. What is the area of this sloping square?

Puzzle of the Week #454 - Heptagon Circle Maximisation

Cut out a heptagonal ‘bite’ from a circle. The bite is four sides of a regular heptagon. The remaining arc length should be constant, and you wish to maximise the area. Below are two figures towards the extremes: a small circle with a small heptagon bite, or a far larger circle, but where the heptagon takes most of that area. In between these two figures there will be a point where the area is at a maximum, but how to construct it?

I had previously solved a far simpler version of this, where the bite was formed by two edges of a square. The optimum in that situation was for the inner vertex of the square to lie on the centre of the circle. In other words the resulting figure would be a three-quarter-circle. It turned out on further examination that the fact that the two lines of the square met at right angles was not important, but the fact that the lines were equal WAS. So if the bite was formed of two identical lines meeting at ANY angle, the maximum area for that angle is achieved when the vertex lies on the centre of the circle:

 How can you use this information to construct the maximum area in the case of the heptagon?

Puzzle of the Week #453 - The Long Game

I have 7 copies each of 14 cards labelled A-N. Each day I try to play a card. I cannot play the same letter as the previous day. In addition if I played a letter for the nth time, the next day I cannot play another letter for the nth time. If I cannot play a card I must take a day off, but the following day I can play whatever card I want.

How many days will it take me to play all of the cards?

Puzzle of the Week #447 - New Order

The following list of words are in alphabetical order. Change them into a different order that also has some reason behind it but has nothing to do with the alphabet.

 

Antenna

Freighter

Lioness

Network

Zaniness

 

  

****Thank you for trying to solve my puzzles this year. This is the last for 2023 as I take a break over the Christmas period. I’ll be back in the new year.****

Puzzle of the Week #445 - Functional Equation

2*f(3/(3-x)) – f(x) = 135*x

 What is the value of f(4)?

 f(x) is a function of x. The above is a ‘functional equation’, where f() of more than one different value appears as an unknown. It may look intimidating, but if you approach it in the right way, it is solvable without any advanced mathematics. As a slight push in the right direction, see what happens if you let x=4.