Puzzle of the Week #92 - Nought and Crosses

This isn't a puzzle in any traditional sense, however I thought I'd share it here anyway:

Noughts and crosses (tic tac toe) is famously a game which will always end in a draw if both players are playing rationally.

However with a simple change to the rules which would appear to be both sensible and fair - namely that neither player may place their first mark in the centre square - the player who goes first can always force a win.

The puzzle then, if there is one, is merely for the reader to satisfy them self that this is true, and to determine, if you were going first, how you would force a win.

 

Puzzle of the Week #91 - Quotebreaker

I have taken a quotation, and I have replaced each of the letters with one-, two- of three-digit numbers according to the table below. Can you change it back to letters?

Be careful though, 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).

This particular quotation is from Leo Tolstoy.

1 31132 3332 1 110203311110313210 31213011 11313023 201103 11033 12331021311110 2021103 1333130 13210 1031121 11033 202131103113012 1111211102121 313310232213213, '11033101121 21'31 1333213213 11033 33311211102 1101131132110121-122111211 31213011103 13210 110201132 10211103110 111100 13210 103301111100'.

Puzzle of the Week #86 - Paddocks

Draw fences between some of the posts so that each post is at the junction of exactly three fences.

These fences will divide the field into several paddocks; any paddock whose area is greater than a single triangle will contain a number, which will indicate the area of the paddock that contains it.

The boundary fence is already in place, so any post on the boundary only needs one more fence emerging from it in order to make up its full complement.

For example:

Here is the puzzle:

Puzzle of the Week #81 - Fixture List

In a new football league there are five teams: Allesley, Bannerbrook, Canley, Dunsmore and Earlsdon. Over the course of 10 weeks, each team must play every other team twice (once home and once away). There will be two games each week, with one team getting a rest. It is your job to organise the fixtures.

Is it possible to organise the fixtures so that each team alternates between home and away games?

If it is, find an example fixture list; if it is not, why not?

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?