Puzzle of the Week #79 - Cumulative Unrepetitiousness

This is really quite a tough one, so I wouldn’t blame you for sitting this one out. I promise to be gentler next week!

I have invented a function of a number, let’s call it the ‘cumulative unrepetitiousness’, C(n). The function looks at all the positive whole numbers up to and including n, and splits them into two categories: into category A go all of the numbers which contain some identical consecutive digits (such as 113 or 3457335), and into category B go all of the numbers that contain no identical consecutive digits (such as 34567 of 2323). C(n) is the size of category B minus the size of category A.

The value of C(n) either goes up or down by 1 as n goes up by 1:

C(1) = 1

C(2) = 2

C(3) = 3

C(4) = 4

C(5) = 5

C(6) = 6

C(7) = 7

C(8) = 8

C(9) = 9

C(10) = 10

C(11) = 9

C(12) = 10

etc.

As you can see, for small values of n, C(n) is always positive.

When n gets big enough, C(n) is always negative.

For a brief time in the middle, C(n) crosses the zero line several times. In fact there are a total of 35 positive values of n for which C(n) = 0, before it heads off into the negative zone for ever more.

Your task is simply to find the first of these.

 

Puzzle of the Week #78 - Contract Confusion

At my office there is often confusion created around our five-digit contract numbers. A common error involved numbers with repeated consecutive digits, for example 54467 might be wrongly written as 54667. To combat this problem it was decided that, for the contract numbers beginning with 6, no repeated consecutive digits would be allowed. So 60000 is out straight away. In fact the first acceptable number of this new scheme would be 60101.

How many of the 10000 possible numbers from 60000 to 69999 can still be used as contract numbers?

Puzzle of the Week #77 - Bed of Nails

I have a bed of nails where the nails, of equal height, are arranged in a square array with 6cm between adjacent rows and 6cm between adjacent columns. I have a globe, 11cm in diameter, which I place onto the bed of nails. It settles so that it is resting on the tips of four nails.

How much higher will the top of the globe be than the tips of the nails?

Puzzle of the Week #74 - Two Coins

In the country of Brookgladia, there are only two denominations of coinage, both whole number amounts higher than 2. Using just the two values of coins it is possible to total any whole number amount from 64 upwards, however it is impossible to total 63.

 

What are the values of the two coins?

Puzzle of the Week #73 - Walk in the Park

There is a circular park with a radius of 200m. I want to build a circular lake of radius 91m inside the park.

I wish to be able to walk from a point on the edge of the park, on a straight line tangential to the lake, until I meet the edge of the park again, and then do the same twice more and end up where I started.

If I place the lake centrally in the park, I will go past my original point on the circumference.

If I place it too far from the centre of the park, I will not get all the way round to my starting point.

How far from the centre of the park should the centre of the lake be to ensure that I end up precisely where I started?

Puzzle of the Week #72 - Ten Pin Bowling

This game uses all 100 scrabble tiles drawn at random, including the two blanks (which can represent any letter of your choosing).

The scoring system is the same as that of real ten-pin bowling: you get points for each word, equal to how many letters in the word.

In addition, there are bonus points available as follows:

If you get a spare (use all letters in one frame using two words), you get bonus points equal to the next word you score.

If you get a strike (a ten letter word), you get bonus points equal to the next two words you score.

If you only get one word in a frame, and it's not a strike, then for the purposes of bonus points, you get a zero length word too.

In real tenpin bowling, if you get a strike or a spare on the tenth frame, you get an eleventh frame to determine your bonus points, and if you were lucky enough to get a strike on the tenth and eleventh frames, you would get a twelfth frame.

In this game, there are no eleventh and twelfth frame, so to determine any bonus points you are entitled to after the tenth frame, look back at the words you scored in the first and second frames.

Theoretical maximum points is 300 (for 10 strikes), but anything over 100 is respectable. I’ve no idea what is the most points achievable with this selection, and will be playing along with everyone else!

Puzzle of the Week #70 - Divisibility Test

It is easy to test a number for divisibility by 2, 5 or 10, by just looking at the final digit. Divisibility by 3 or 9 is almost as easy, whereby you add together the digits of your number, and if the resulting total (which will necessarily be smaller than your original number) is divisible by 3 (or 9), then so was your original number.

But can you devise a test for divisibility by 7, 11 or 13 (the same procedure for all three) where you can very simply, using addition and subtraction, reduce a number of however many digits, down to a three digit number, which will be divisible by 7, 11 or 13, if and only if your original number was?

Puzzle of the Week #67 - Ten Pin Bowling

This game uses all 100 scrabble tiles drawn at random, including the two blanks (which can represent any letter of your choosing).

The scoring system is the same as that of real ten-pin bowling: you get points for each word, equal to how many letters in the word.

In addition, there are bonus points available as follows:

If you get a spare (use all letters in one frame using two words), you get bonus points equal to the next word you score.

If you get a strike (a ten letter word), you get bonus points equal to the next two words you score.

If you only get one word in a frame, and it's not a strike, then for the purposes of bonus points, you get a zero length word too.

In real tenpin bowling, if you get a strike or a spare on the tenth frame, you get an eleventh frame to determine your bonus points, and if you were lucky enough to get a strike on the tenth and eleventh frames, you would get a twelfth frame.

In this game, there are no eleventh and twelfth frame, so to determine any bonus points you are entitled to after the tenth frame, look back at the words you scored in the first and second frames.

Theoretical maximum points is 300 (for 10 strikes), but anything over 100 is respectable. I’ve no idea what is the most points achievable with this selection, and will be playing along with everyone else!

 

 

 

 

 

Puzzle of the Week #62 - Tenpin Bowling

The object of this game is to try to score as highly as possible by using letters in a given rack to form words. Just like in real ten pin bowling, you have a maximum of two attempts at each rack.

If you get a ten-letter word, that is a Strike and is worth 20 points.

If you find two words that between them use each of the ten pins once each, that is a Spare and is worth 15 points.

Any fewer than that, just total up the letters used to give your point total. So if you find a five-letter word and a three-letter word, that rack will have scored you 8 points.

I have randomly generated the racks by drawing 50 scrabble tiles out, discarding only the blanks (unlike the previous outing of this puzzle, where I tweaked it so that at least a spare was achievable on each rack - this way I can legitimately take part myself).

What is the highest total you can achieve over the five racks?

Puzzle of the Week #61 - Power Play

61 raised to the power of 61

or 61 61’s all multiplied together,

or 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61 x 61

is a very large number, with 109 digits altogether.

What are the last three digits?