Puzzle of the Week #470 - Word List

The following words all have something specific in common:

SCAR   THOU  NAPA  SOLE   TECH  DOLL   DIVE    DIVE    GALA

The order they are in is related to the thing they have in common.

DIVE is in the list twice – this is not a mistake.

 Which one of the following words also belongs in the list, and where in the list should it be placed?

 BECK   CLEF   PARA   PAST

Puzzle of the Week #466 - Perimeter

I’ve started with a 3x3 square, and divided it into 9 1x1 squares as shown. I’ve added four 1x1 squares such that for each of them one corner lies on a corner of the central 1x1 square, and the corner opposite that lies outside of the 3x3 square. I’ve used this as the basis of the lower figure, a bizarre 20 sided polygon.

What is the perimeter of this polygon?

Puzzle of the Week #464 - Mystery Word Game

A player plays a Wordle type game, where feedback is given after each guess. The player correctly finds the answer ‘DELAY’ on her fourth guess. Her guesses were:

 I  S S U E

A L A R M

B I D E T

D E L A Y

At each point the guess could have been the solution, in other words she hasn’t employed any clever elimination tactics. There were still a handful of other possibilities at the point where she got the answer, but it was not unreasonable that she guessed it after four guesses given the feedback she had.

What feedback was she receiving after each guess?

An extra challenge if you fancy it: find another word that still might have been the answer after the first three guesses, with the feedback as it would have been for DELAY.

 

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?