Puzzle of the Week #441 - Puzzle 441

It is possible to write the numbers 1 to 441 in a particular order such that the sum of every consecutive pair of numbers is either 321, 442, or 763. For instance, 400 might be followed by 42 or 363.

When you do this, what numbers are at the start and the end of the sequence?

And for bonus points, what number would be in the exact centre of the sequence?

Puzzle of the Week #438 - parkrun Density

I have 17 parkruns within 17 miles from my house.

Assuming the density is broadly uniform, if I calculate the same stat but in kilometres, such that I have n parkruns within an n kilometre radius, what is a good estimate for n?

(There are ~1.61km in a mile).

 a)    11

b)    17

c)    22

d)    27

e)    44

 

Puzzle of the Week #437 - Rugby Scores part 2

We saw last week how the number of ways of reaching a particular score by adding 3s, 5s and 7s can be found using the recurrence relation:

a(0)=1, a(n)=a(n-7)+a(n-5)+a(n-3) for n not equal to 0.

Now we go slightly abstract and just look at, for all the values of a(n) for n greater than or equal to 0, the parity (odd or even) of each value.

I’ll give a simpler example just to be clear. If a(n)=a(n-2)+a(n-1) and a(0)=1, you get this familiar sequence: 1,1,2,3,5,8,13,21,34,55, etc. Just looking at the parity we get: OOEOOEOOEO etc. It seems to be the case that two-thirds of the numbers in that sequence are odd, and indeed that is the case.

The rugby sequence begins 1,0,0,1,0,1,1,1,2,1,3,3,3,6,5,8,10,11,17…

Which is: OEEOEOOOEOOOOEOEEOO etc

If you continue that sequence forever, what proportion of the values are odd?

Puzzle of the Week #436 - Rugby Scores

As the Rugby World Cup is currently ongoing I thought I’d do a puzzle based on the scoring system. A kick (drop goal or penalty) is worth 3 points, an unconverted try is worth 5 points, and a converted try is worth 7 points.

Some scores are impossible to reach: you can’t get 1, 2 or 4.

8 is the first score that’s achievable in two ways, as long as we say that ‘3 then 5’ is different from ‘5 then 3’.

In how many ways is the score ‘23’ achievable?

Puzzle of the Week #434 - Four Guesses

I think of a number in the range 1 to 4. You try to guess what it is.

Assuming your guess was incorrect I must now change my number to either one more or one less than my previous number (but still within the range 1 to 4). You try to guess again.

We continue like this, with me changing to an adjacent number and you guessing what my current number might be.

What strategy could you come up with that will always successfully guess my number within 4 guesses?

Puzzle of the Week #432 - Ace of Base

What is the maximum value of a number whose digits add up to 10, if you are allowed to decide the base of the number.

I have to put in place a couple of rules:

You can’t use any zeroes. Clearly you could make a number arbitrarily large by adding a string of zeroes at the end.

The first digit must be just one less than the base you are using, otherwise you could just have, say, the number 55 but say it was in an arbitrarily large base.

The decimal number 91 and the binary number 1111111111 both follow these rules, however they are not the maximum. What is?

Puzzle of the Week #430 - 1x1x1 Rubik's Cube

1x1x1 Rubik’s cubes do exist, but since they have no moving parts and therefore cannot be scrambled, they are really just novelty joke objects. What I’m imagining here I don’t believe exists in the real world, and I’m not sure what the mechanism would be, but with my cube it is possible to swap the colours of two adjacent faces. What I’ve shown below is the solved state, which is that red and orange are opposite, blue and green are opposite, white and yellow are opposite, and red-white-blue run clockwise around their shared vertex. Any whole-cube rotation of this is still the solved state, but a mirror reflection is not. Obviously to solve this cube is trivially easy, so the question is, what is the minimum number of moves (adjacent colour swaps) by which it is ALWAYS possible to solve the cube, no matter what the starting scramble?

Puzzle of the Week #429 - Strange Stamps

The islanders of Fictitia have a rather eccentric postal system. Postage for an item can be ANY whole number amount from 1 dinar to 26 dinari, and you MUST use exact postage.

Frustratingly, there is only space on the envelopes in Fictitia to attach a maximum of TWO stamps.

What is more, they only have SEVEN different denominations of stamps, can you work out what they are?

Puzzle of the Week #428 - Doubly Generated Hexagon

Starting with a 3,4,5 triangle, draw a square on each side.

Connect the outer corners of these squares to form a hexagon.

Draw a square on three sides of the hexagon as shown.

Finally connect the outer corners of those squares to form a larger hexagon.

What is the area of the largest hexagon?

As a follow up, if you fancy it, what if the initial right-angled triangle, instead of having legs 3 and 4, it had legs a and b. What is the area of the outer hexagon in terms of a and b?

Puzzle of the Week #427 - House Numbers part 2

My two friends live on the same street. Oddy lives on the odd side of the street, and Evelyn lives on the even side.

Just looking at the houses on the odd side, the sum of all the houses below Oddy’s house (1+3+5 etc) is exactly a quarter of the sum of the house numbers above Oddy’s house on the odd side.

Looking at the even side, the sum of the house numbers below Evelyn’s house (2+4+6 etc) is exactly double the sum of the house numbers above Evelyn’s house on the even side.

How many houses are there on Oddy’s and Evelyn’s street?

Puzzle of the Week #426 - House Numbers

I live on the odd side of quite a long street. My house number has three digits.

The sum of all the house numbers to the left of my house (1+3+5+7+9+11+etc) is the same as the sum of the house numbers to the right of my house all the way to the end of the road.

What number house do I live at?

Puzzle of the Week #424 - Sock Drawer

Eight pairs of socks have been placed in a drawer such that the pattern they make has 4x rotational symmetry. Identify where each of the socks go and discover which two socks from the same pair are touching.

Example

With four pairs of smaller socks, where the ‘1’ pair is touching:

The Puzzle: