Puzzle of the Week #313 - Alien Number System

Imagine a number system where the only numbers are those that are 1 greater than a multiple of 20, for instance, 21, 81, 1741. You cannot add or subtract using this number system without the result being a number outside the number system, however it is possible to multiply, as multiplying together two numbers that are each 1 greater than a multiple of 20 will result in a third number that is also 1 greater than a multiple of 20. For example, 21 x 61 = 1281.

‘Prime’ numbers exist in this system, defined as numbers that cannot be formed by multiplying together two smaller numbers in the number system. All actual primes, such as 41, are obviously still prime in this system, but other numbers, such as 21 or 81, are not prime usually, but are in this system.

One well known fact about ordinary numbers is that they are the product of prime numbers in exactly one way, for example 72 = 2x2x2x3x3. However, it is possible for numbers in this special number system to be the product of ‘prime’ (within the system) numbers in more than one way.

What is the smallest number in this number system that is the product of ‘primes’ in two distinct ways?

Puzzle of the Week #312 - Wolf Tooth

A Wolf Tooth cube is a strange and interesting puzzle. It is almost like solving two puzzles at once. In essence it is a cube and an octahedron intersected. Each of the six cube faces has one of the octahedron vertices in the centre, a square based pyramid with four different colours on it. Each of the eight octahedron faces has one of the cube vertices in the centre, a triangular based pyramid with three different colours on it.

 

The arrangement of colours on the cube part are as follows:

Red is opposite Orange

White is opposite Yellow

Blue is opposite Green

Red White and Blue appear clockwise on their shared vertex

 

The arrangement of colours on the octahedron part are as follows:

Red is opposite White

Yellow is opposite Silver

Purple is opposite Blue

Orange is opposite Green

Red Yellow Purple and Orange appear clockwise around their shared vertex

 

It is possible to orient the octahedron through the cube such that none of the same colours on the cube and octahedron are in contact?

If so what four colours appear on the octahedron vertex in the middle of the green cube face?


lanlan-wolf-tooth.jpg

To illustrate the objective of the puzzle, in the cube above the white cube face and the white octahedron face are not in contact, whereas the green cube face and the green octahedron face are in contact, which is not permitted within this puzzle.

(For the purposes of this puzzle I have changed the order of the colours on the octahedron part from the colouring on an actual Wolf Tooth cube, shown here.)

Puzzle of the Week #311 - Diamond Ring

ABCD is a square with side length of 2, rotated through 45 degrees so that the diagonal BD becomes horizontal. A circle of unknown radius is drawn through B and D as shown. Within the large circle two further circles are drawn, respectively above and below BD, and the maximum size they can be whilst staying within the large circle.

The region that lies within the large circle but outside of the two smaller circles is shaded.

What is the area of the shaded region?

diamond ring puz.JPG

Puzzle of the Week #310 - Always True?

I happened to notice that if I took a number that was the sum of two different squares (eg, 4+1 = 5) and multiplied it by a different number that was also the sum of two different squares (eg, 9+1 = 10), the result (50) would also be the sum of two different squares: (49+1).

But is this always true?

Puzzle of the Week #308 - Shadowbox

Here is a puzzle you might not have met before, as it’s one of my own invention. A cross between a Fill-In and a Skeleton which I call ‘Shadowbox’.

Place all the listed words into the grid, crossword style, such that every white square contains a consonant, and every grey square either contains a vowel (A, E, I, O, U), or becomes a black square. The pattern of black squares in the grid is fully symmetrical.

shadowbox puz.JPG

If you enjoy this, I wrote an entire book of them, available online:

https://www.amazon.co.uk/Shadowbox-Logical-Crossword-Puzzles-Elliott/dp/1447861965

 

Puzzle of the Week #307 - Descend the Ladder

This puzzle is based on a similar algorithm to last week’s puzzle, but this time the sequence descends from any composite number, and stops when it reaches a number that is not composite (so, either a prime number or the number 1).

At each stage find the largest prime factor of your number and SUBTRACT this plus one from your number to get the next number, for instance: 24 -> 20 -> 14 -> 6 -> 2

There are many, possibly infinitely many, starting numbers that terminate at either 1 or 2, whereas there are no starting numbers that reach 3 or 7. 5 however is a more interesting case, there is a relatively small set of numbers that lead to 5. Can you find them all?

Puzzle of the Week #306 - Climb the Ladder

A while ago I ‘invented’ an interesting mathematical algorithm as follows.

 Start with any whole number ‘n’.

If ‘n’ is prime, stop.

If ‘n’ is composite, list its prime factors and find the largest, let’s call it ‘m’

Let your new ‘n’ be equal to n+m+1, and repeat the whole sequence.

 For instance if we start with 15:

 15

Not prime, highest prime factor is 5, so add 6

21

Not prime, highest prime factor is 7, so add 8

29

Prime, stop

If you start with the number 38, which prime number do you eventually end up at? 

Puzzle of the Week #304 - Compass Points

Four lines of lengths 12, 31, 27 and 10 are drawn respectively from the North, East, South and West points on a circle, heading directly away from the circle’s centre as shown.

compass points puz.JPG

 The four endpoints are joined with straight lines to form an irregular quadrilateral. Two opposite angles of this quadrilateral, A and B, add to 180 degrees.

 What is the radius, r, of the circle? 

(non)Puzzle of the Week

As you may know, whilst these days I publish a solution each week a few days after the puzzle, for a few years this wasn’t the case. From 2019 and earlier, most of the puzzles are missing a solution. If you want, you can help!

The following link drops you much earlier in the puzzle feed, but you can scroll forward and backwards through the hundreds of puzzles:

http://www.elliottline.com/puzzles-1?offset=1485000000000

Your mission, should you choose to accept it, is to choose a puzzle that doesn’t already have a solution (there are plenty to choose from), and write one. By which I mean, not merely giving the answer, but also an explanation, a way of arriving at the solution, or a rationale for why the answer is what it is. Feel free to use diagrams.

I intend to then take your solution and publish it, and I will of course give you full credit. Send your solution to me at ell.ell@talk21.com

Good luck, and thank you in advance!

Puzzle of the Week #300 - Winner Stays On

Four friends, Alfie, Billie, Charlie and Debbie, play a series of games on their pool table. At each point, two of the friends are playing each other while the other two are reduced to spectating. After each game, the winner stays at the table and will go on to play whichever of the two spectators has been waiting the longest since their last game, and the loser becomes a spectator for the next game, in order to ensure everybody gets to play.

 

After they have finished Alfie has played in eight of the games, Billie three, Charlie six and Debbie five.

 

Who lost in the ninth game?

Puzzle of the Week #299 - Change a Letter

Change one letter from each word, and THEN re-space to form an aphorism:

 

For example:

TO  YOGA  INFO  ILL  GRIN  FAN  LIE  TIER

becomes:

TR  YAGA  INFA  ILA  GAIN  FAI  LBE  TTER

then:

TRY AGAIN FAIL AGAIN FAIL BETTER

 

 

CHEF  CAME  TO  IT  BUT  NOT  WISE  AT  BRAG  AT  OUR  ASH  AS  FAST  ONE

Puzzle of the Week #298 - Unit Fractions

It is well known that if you add together all of the unit fractions, 1 + 1/2 + 1/3 + 1/4 + 1/5 + … all the way to infinity, the answer is also infinity (although it approaches it ridiculously slowly).

However, if we throw out any that have a never-ending decimal, such as 1/3 (0.333…), 1/6 (0.166…) ,1/7 (0.142857…) etc, and only include those that have a terminating decimal expansion:

S = 1 + 1/2 (0.5) + 1/4 (0.25) + 1/5 (0.2) +1/8 (0.125) + 1/10 (0.1)… all the way to infinity,

we do get an actual number as the result. What is it?


Puzzle of the Week #295 - Base 4 Code

I have taken a quotation, and I have replaced each of the letters with the numbers that denote their position in the alphabet. However, I have used the base 4 number system. 

Be careful, as some sequences of numbers could lead to several words, for instance 31110 could mean CAT (3,1,110), but could equally mean MAD (31,1,10). 

base4.JPG

11020213213103  1133310223  33111110  211103110  1233102  110203310311  1132033  3112311  1102011  211103110  3312  1102011  1131121  11020213213103  1133310223  33111110.

22332032  1133333101132.