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Consider a list of all of the three digit numbers that:
are a multiple of 4
contain three different digits
contain at least one odd digit
don’t contain a zero.
Prove that for every number that satisfies these criteria there is exactly one other that uses the same three digits in a different order and also satisfies the criteria.
I have in mind a number, that is the smallest non-zero number that:
when doubled it is a square number,
when tripled it is a cube number,
when multiplied by 5 it is a power of 5, and
when multiplied by 7 it is a power of 7.
How many zeroes does it have after the last non-zero digit?
Although this puzzle might involve very large numbers, it can be approached without a computer or even a calculator.
Similar to last week, but now you have to arrange the numbers 1-15 in a triangle such that each number is the difference of the two numbers immediately below it.
This time there is only one possible solution (plus its reflection), but even if you cleverly narrow down the possibilities with a couple of quick observations, there are still over 60000 triangles to check without a computer.
Alternatively, if I were to give you the bottom row, completing the grid will be trivially easy.
In an effort to find a sweet spot, I have given you the numbers on the left hand slope. Some more numbers can be placed straight away, whereas others you can only initially narrow it down to a couple of possibilities.
Can you arrange the numbers 1 – 10 in this triangle such that each number is the difference of the two numbers immediately below it?
There are in fact four possible solutions (not counting reflections), so extra kudos if you manage to find them all!
Presumably you’re familiar with the online game of Wordle but here’s a brief recap: you try to guess a solution word. If your guess word has any correct letters in the correct place they are coloured green. If your guess has correct letters but not in the correct place these are coloured yellow. If the solution word only has one E for instance, but your guess has two Es and neither are in the correct place, only the first will be coloured yellow.
With that covered, I have come up with a variation. I have selected three English cities, and treated each of them as both the solution word and the guess word in all possible combinations, and coloured them according to Wordle rules. So for instance cities 2 and 3 start with the same letter, and cities 1 and 3 share a middle letter.
Your task is to identify the cities.
You have a well-shuffled pack of 100 cards, numbered 1-100. You draw five cards in turn. Given that the first card you drew was numbered ‘34’, what is the probability that the five cards are in ascending order?
Fred and Sally’s combined age is 65.
Fred is three times as old as Sally was when Fred was the age that Sally is now.
How old are they?
Fill in all the empty squares of the grid with the digits 1-9, such that:
1: A diamond shape denotes that the four numbers adjacent to it add to 20.
2: Every diamond uses a different way to add to 20, so if for example one diamond is surrounded by 1,2,8,9, no other diamonds can use those same four numbers, not even in a different order.
3: Identical digits are not allowed to occupy adjacent squares of the grid, not even diagonally adjacent.
What is the radius of this circle?
Use the following letter pairs to form a series of words: a 4-letter word, a 6-letter word, an 8-letter word, a 10-letter word and a 12-letter word.
Each of the letter pairs can only be used once, EXCEPT for five of them.
Those five special letter pairs appear at the end of each of the five words, and also earlier in the same word. For instance if the 8-letter word ends in OR, it would need to be either:
(OR----OR), (--OR--OR) or (----OROR).
I have a pair of three-digit numbers that are the reverse of one another, let’s call them abc and cba. The greatest common divisor of abc and cba is 7.
What are the two numbers?
What is the highest number that:
is always a multiple of as long as a and b are integers?
In this figure there are two identical large squares and two identical smaller squares. What is the area of the red triangle in terms of the areas of the squares?
In the above continued fraction, replace the letters a,b,c,d with the numbers 1,4,5,7 in whatever order you wish.
What is the maximum value of x?
The following words all have something specific in common:
SCAR THOU NAPA SOLE TECH DOLL DIVE DIVE GALA
The order they are in is related to the thing they have in common.
DIVE is in the list twice – this is not a mistake.
Which one of the following words also belongs in the list, and where in the list should it be placed?
BECK CLEF PARA PAST
If 32 degrees Fahrenheit = 0 degrees Celsius,
and -40 degrees Fahrenheit = -40 degrees Celsius,
calculate in your head what temperature will be the exact same amount of degrees for each but positive for Fahrenheit and negative for Celsius.
Some quarter circles (and semi circles) are arranged in (and out) of a rectangle as shown. What is the length of the line marked with a question mark?
What number is missing from this sequence?
11, 236, 315, 4384, 5175, ???, 7735, 8128, 9135,
I’ve started with a 3x3 square, and divided it into 9 1x1 squares as shown. I’ve added four 1x1 squares such that for each of them one corner lies on a corner of the central 1x1 square, and the corner opposite that lies outside of the 3x3 square. I’ve used this as the basis of the lower figure, a bizarre 20 sided polygon.
What is the perimeter of this polygon?
A straight line is drawn through a circular array as shown. The line is tangent to the lower left and upper right circles. This line passes through some of the other circles and also passes through the space between the circles.
What proportion of the line is within the circles?
