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I have a number, let’s call it n. n doesn’t have any repeated prime factors. For every prime number p, p divides into n IF AND ONLY IF p-1 also divides into n.
For instance, if 23 divides into n, then 22 does also, and if 59 doesn’t divide into n, then neither does 58.
What is the value of n?
What are the missing digits in this product of fractions?
A rectangle ABCD has width 64 and height 80. A right-angled triangle AEF is drawn as shown. What is the minimum that DF could be?
w + x + y + z = 2
w^2 + x^2 + y^2 + z^2 = 2
w^3 + x^3 + y^3 + z^3 = -4
w^4 + x^4 + y^4 + z^4 = -6
What is the value of: w^7 + x^7 + y^7 + z^7 ?
What is the lowest number n such that if a is the highest prime factor of n, then b (not equal to a) is the highest prime factor of (n-a), and c (not equal to a or b) is the highest prime factor of (n-a-b)?
For instance, a higher example is 700. The highest prime factor of 700 is 7, the highest prime factor of 693 is 11, and the highest prime factor of 682 is 31.
There are two ships, a fast one and a slower one, each travel consistently at their own fixed speed.
If the fast ship travels in a straight line from Duxmouth to Boxcote and the slow ship travels from Axton to Caxcombe, and they each set off at the same time, they will collide where the routes cross.
Similarly, if the fast ship travels in a straight line from Caxcombe to Axton and the slow ship travels from Boxcote to Duxmouth, and they each set off at the same time, they will again collide where the routes cross.
If the fast ship travels from Caxcombe to Duxmouth and the slow ship from Axton to Boxcote, and they set off at the same time, clearly they won’t clash, as the routes don’t cross, but which ship will reach its destination first?
It's been a while since I invented a new puzzle type, although this one is a combination of a couple of ideas already around in puzzle world.
Your task is to subdivide the grid into regions, such that every region has 180-degree rotation symmetry. (They might also have 90-degree symmetry and/or reflective symmetry, but this is not necessary).
Every square that in the solved grid is bordered by 2 or fewer lines, is denoted by the number of lines bordering it. Any square that will have 3 or 4 lines bordering it, is left blank.
The only other rule is that you can't have two single square regions next to each other, hence why the top right corner of the example must be a pair.
In this figure, the shaded area is in between an isosceles triangle with sides 1, 1 and x, and an equilateral triangle with sides x.
If x is close to zero, the isosceles triangle becomes very thin and the shaded area in very small.
If x is close to 1, there is very little difference between the isosceles triangle and the equilateral triangle, and again the shaded area is very small.
In between, the shaded area is at a maximum. For what value of x is this the case?
Usually I like to engineer these puzzles so that the solution is a whole number, however this is not possible in this case, but the exact solution can be expressed quite concisely. A numerical solution is also acceptable to, let’s say, four decimal places.
The longest streak of consecutive numbers, NONE of whose digit sums is a multiple of 7, is 12 in a row. For instance:
994, digit sum = 9+9+4 = 22, remainder after division by 7 = 1
995, digit sum = 9+9+5 = 23, remainder after division by 7 = 2
996, digit sum = 24, remainder after division by 7 = 3
997, digit sum = 25, remainder after division by 7 = 4
998, digit sum = 26, remainder after division by 7 = 5
999, digit sum = 27, remainder after division by 7 = 6
1000, digit sum = 1, remainder after division by 7 = 1
1001, digit sum = 2, remainder after division by 7 = 2
1002, digit sum = 3, remainder after division by 7 = 3
1003, digit sum = 4, remainder after division by 7 = 4
1004, digit sum = 5, remainder after division by 7 = 5
1005, digit sum = 6, remainder after division by 7 = 6
By the same notion, the length of the longest streak of consecutive numbers, NONE of whose digital sums is a multiple of 13, happens to be an exact multiple of 13 itself.
How long is the streak, and can you find an example?
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?
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?
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.)
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?
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?
I have three straight lengths of fencing, measuring 25m, 33m and 39m, and a long straight brick wall. Using the wall and the three fences, what is the greatest area I can enclose?
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.
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
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?
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
What are the last five digits of the following number?
To be clear, in the absence of brackets, a power tower in calculated from the top, so that 3^3^3 = 3^27, not 27^3.
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.
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?
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!