Puzzle of the Week #430 - 1x1x1 Rubik's Cube

1x1x1 Rubik’s cubes do exist, but since they have no moving parts and therefore cannot be scrambled, they are really just novelty joke objects. What I’m imagining here I don’t believe exists in the real world, and I’m not sure what the mechanism would be, but with my cube it is possible to swap the colours of two adjacent faces. What I’ve shown below is the solved state, which is that red and orange are opposite, blue and green are opposite, white and yellow are opposite, and red-white-blue run clockwise around their shared vertex. Any whole-cube rotation of this is still the solved state, but a mirror reflection is not. Obviously to solve this cube is trivially easy, so the question is, what is the minimum number of moves (adjacent colour swaps) by which it is ALWAYS possible to solve the cube, no matter what the starting scramble?

Puzzle of the Week #429 - Strange Stamps

The islanders of Fictitia have a rather eccentric postal system. Postage for an item can be ANY whole number amount from 1 dinar to 26 dinari, and you MUST use exact postage.

Frustratingly, there is only space on the envelopes in Fictitia to attach a maximum of TWO stamps.

What is more, they only have SEVEN different denominations of stamps, can you work out what they are?

Puzzle of the Week #428 - Doubly Generated Hexagon

Starting with a 3,4,5 triangle, draw a square on each side.

Connect the outer corners of these squares to form a hexagon.

Draw a square on three sides of the hexagon as shown.

Finally connect the outer corners of those squares to form a larger hexagon.

What is the area of the largest hexagon?

As a follow up, if you fancy it, what if the initial right-angled triangle, instead of having legs 3 and 4, it had legs a and b. What is the area of the outer hexagon in terms of a and b?

Puzzle of the Week #427 - House Numbers part 2

My two friends live on the same street. Oddy lives on the odd side of the street, and Evelyn lives on the even side.

Just looking at the houses on the odd side, the sum of all the houses below Oddy’s house (1+3+5 etc) is exactly a quarter of the sum of the house numbers above Oddy’s house on the odd side.

Looking at the even side, the sum of the house numbers below Evelyn’s house (2+4+6 etc) is exactly double the sum of the house numbers above Evelyn’s house on the even side.

How many houses are there on Oddy’s and Evelyn’s street?

Puzzle of the Week #426 - House Numbers

I live on the odd side of quite a long street. My house number has three digits.

The sum of all the house numbers to the left of my house (1+3+5+7+9+11+etc) is the same as the sum of the house numbers to the right of my house all the way to the end of the road.

What number house do I live at?

Puzzle of the Week #424 - Sock Drawer

Eight pairs of socks have been placed in a drawer such that the pattern they make has 4x rotational symmetry. Identify where each of the socks go and discover which two socks from the same pair are touching.

Example

With four pairs of smaller socks, where the ‘1’ pair is touching:

The Puzzle:

Puzzle of the Week #422 - Boggle E

Find all of these common words within this Boggle grid, each containing the letter E.

Each letter after the first must be a horizontal, vertical, or diagonal neighbor of the one before it.

No individual letter cube may be used more than once in a word.

B - - - E

B E - -

B E - -

C - - E

C - - E

C - - - - - E -

D E - -

D E - -

F - - E

F - - E -

F - - E

G E - - -

H E - -

H E - -

L - E -

L - - E

M - - E

 

 

 

Puzzle of the Week #421 - Odd Prime Balance

I thought about throwing you in the deep end on this one, but even just the last part of the puzzle is challenging enough, so I’ll lead you up to that point:

We saw from the Prime Balance puzzle (POTW #380) that if we want a number of distinct prime number blocks equally spaced around a circle to have a centre of gravity at the centre of the circle, the least number of blocks is six. This is because if you want distinct rational weights, the number of weights cannot be a prime or a prime power. If we add in a further stipulation that the number of blocks must be odd, the lowest number that is odd but neither a prime nor a prime power is 15.

Using the fact that 15 is 3 x 5 we can set up a system of equations as we did before:

 

A = V

B = W+T

C = X+U

D = Y

E = Z+T

F = V+U

G = W

H = X+T

I = Y+U

J = Z

K = V+T

L = W+U

M = X

N = Y+T

O = Z+U

 

Whatever we choose for T to Z the system of blocks will be balanced around the centre of the circle. We still have to ensure that the numbers A to O are all prime and all different. What is the least possible total weight of the 15 blocks?

 

Puzzle of the Week #419 - Minimum Triangle

This right-angled triangle is shown with its inscribed circle (incircle). The distances a and b make up the distance from each end of the base to the tangent point of the incircle. c is the height of the triangle, perpendicular to the base.

If I tell you that a, b and c are all whole numbers, and all different, what is the minimum sum of a, b and c?

Puzzle of the Week #418 - Fourth Circle

Three circles of radius 70, 20 and 50 are mutually tangent.

A fourth circle is introduced, which passes through the centre of the 50 circle, and which is also tangent to both the 20 circle and the 70 circle at the point that they are tangent to one another.

What is the radius of this fourth circle?

Puzzle of the Week #417 - Triple Duality

The answer to each clue is a 3-, 6-, or 9-letter word. If you split these into 3-letter chunks, you will find 20 different ‘triples’ each appearing twice.

For example if one answer was ‘bellow’ then ‘bel’ and ‘low’ would appear somewhere else too, for instance in ‘disbelief’ and ‘lowest’, but then ‘dis’, ‘ief’ and ‘est’ would have to appear somewhere else too, etc, until all triples have appeared exactly twice.

 

Age

Be present at

Christian religious leader

Condensed water droplets

Endurable

Finish

Imperfections

'In the name of Allah'

Mold

Opinion, recommendation

Outcast

Part of a circle

Part of speech

Part of triple jump

Set upon

Shiite religious leader

Softcover

Suitable for purpose

Tropical fruit

Underneath the ...

Vanilla ..., rapper

 

Puzzle of the Week #415 - Alchemy

You can change one word into another of the same length by a series of steps whereby at each step you change one of the letters and rearrange to form a valid word. This differs from usual word ladders where you aren’t permitted to rearrange the letters.

LEAD and GOLD both contain L and D so would only require two steps to change between them, so just one intermediate word, for instance: LEAD – GLAD – GOLD.

Can you change CARBON into SILVER in only five steps (four intermediate words)?

Puzzle of the Week #411 - Best Remakes

Watching the Oscars this year I noticed that one of the nominees for Best Picture was a remake of one of the very first winners almost a century ago. In fact, in four of the last five years, a Best Picture nominee has been a remake of a film to previously be nominated for Best Picture. I trawled the lists and found several more examples for you to try and discover below. Where a film has three or more words in the title, I’ve given the initials; where it has only a single or two-word title, I’ve given an anagram.

 

Winner 1930, nominated 2022: AQOTWF

Winner 1961, nominated 2021: WSS

Nominated 1933 and 2019: TIME NOT WELL

Nominated 1937 and 2018: ASIB

Nominated 1935 and 2012: SMELLIER BASES

Nominated 1952 and 2001: GERONIMO ULU

Nominated 1943 and 1978: HCW

Nominated 1936 and 1968: RAJ

Nominated 1934 and 1963: ACE PATROL

Winner 1935, nominated 1962: MOTB