Your Custom Text Here
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!
Three unit radius circles are arranged to each be tangent to the other two. Four lines: AB CD EF GH are drawn through these tangent points as shown, extending both ways to meet the circles again, with AB and EF drawn horizontally and CD and GH drawn vertically. If AB and CD have the same length, what is the length of FG?
Five identical triangles are placed in a row with their bases collinear as shown. A couple of diagonal lines are drawn from the apexes of some triangles to the lower vertices of others. If each of the five triangles has an area of 1, what is the area of the region marked with an A?
A group of 21 friends are seated around a large circular table. By a strange coincidence, the sum of the ages of ANY six consecutively seated friends adds up to 200. If the person at seat 1 is aged 25 and the person and seat 8 is aged 33, how old is the person at seat 15?
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?
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
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?
AB^4 + CD^4 = 2021
AB^3 + CD^3 = 485
AB^2 + CD^2 = 101
AB + CD = 5
What is ABCD?
A few special numbers can be expressed as the product of a set of three or more integers in arithmetic progression. For instance 2x5x8 = 80, 3x4x5x6x7 = 2520, 4x6x8x10 = 1920.
Of the three-digit numbers that starts with 38, TWO are those special numbers.
Which ones, and how?
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).
K, L and M are all positive whole numbers.
For the certain special values of K that we seek, the same values of L and M that cause (KxL)+(4xM) to be a multiple of 11 also cause (KxM)+(5xL) to be a multiple of 11.
For instance, K ISN’T 2, because some values of L and M that make (2xM)+(5xL) a multiple of 11 (eg L=1,M=3) when you plug those same values of L and M into (2xL)+(4xM) give a number that is NOT a multiple of 11 (in this case 14).
Out of the possible values of K for which the divisibility by 11 of (KxL)+(4xM) and (KxM)+(5xL) are always in agreement, what number is the THIRD LOWEST PRIME?
Calculate the value of the following without electronic assistance:
I have taken some quotations, 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).
