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.
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.
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 #303 - Three Tangent Circles
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?
Puzzle of the Week #302 - Five Triangles
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?
Puzzle of the Week #301 - Circular Table
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?
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 #297 - 2021 Puzzle
AB^4 + CD^4 = 2021
AB^3 + CD^3 = 485
AB^2 + CD^2 = 101
AB + CD = 5
What is ABCD?
Puzzle of the Week #296 - Three Hundred and Eighty Something
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?
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).
11020213213103 1133310223 33111110 211103110 1233102 110203310311 1132033 3112311 1102011 211103110 3312 1102011 1131121 11020213213103 1133310223 33111110.
22332032 1133333101132.
Puzzle of the Week #294 - Special K
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?
Puzzle of the Week #293 - Mental Calculation
Calculate the value of the following without electronic assistance:
Puzzle of the Week #292 - Base 4 Code
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).
12133111 3132 1112332110 1021113021110121, 2111110 12133111 3132'110 1112332110 1102011 33332103111011111132311103 3312 1112332110213213 1021113021110121.
Puzzle of the Week #291 - Divisible by 32
If n can be any natural number (positive whole number), when is (26^n + 6^n) NOT divisible by 32?
Puzzle of the Week #290 - Special Sequence
This is a special sequence:
3 1 2 1 3 2
For each number ‘k’ from 1 to 3, the number of numbers between the pair of ‘k’s is equal to k.
In other words, there is one number between the pair of 1s, two numbers between the pair of 2s and three numbers between the pair of 3s.
Can you create a similar sequence containing pairs of numbers but using numbers 4, 5, 6 etc up to n (n can be whatever you need it to be to make a sequence that has no gaps), where for each number k, the number of numbers between the pair of ‘k’s is k (so there will be four numbers between the pairs of 4s, five numbers between the pair of 5s, etc.)?
Puzzle of the Week #289 - Power Play
What is the remainder of (10^101 + 11^101) when divided by 21?
Puzzle of the Week #288 - Round Table
A family that consists of parents and 6 children sit around a table in age order: father, mother, eldest child etc, so the youngest child sits next to the father.
The gender of each of the children is male or female, with 50% probability of each.
Knowing only this information, what is the probability that all the males are seated together?
