Puzzle of the Week #248 - Quartet

The four numbers 2,3,4 and 8 can be combined in pairs in six different ways, and the product of those six pairs will be

2 x 3 = 6

2 x 4 = 8

3 x 4 = 12

2 x 8 = 16

3 x 8 = 24

4 x 8 = 32

The sum of the original four numbers (2, 3, 4 and 8) is 17.

Can you find a different quartet of POSITIVE numbers whose products of pairs are also 6, 8, 12, 16, 24 and 32, but whose sum is less than 17?

Puzzle of the Week #246 - Ghost Crossword

Use logic to fill in the crossword grid given only the clue numbers and the following rules:

1) The crossword is numbered in the usual way.

2) The grid is fully symmetrical.

3) The white area must all be connected together.

4) ‘Words’ are at least three letters long.

5) No 2x2 black squares are allowed.

6) No row or column is entirely white or entirely black.

ghost puz with numbers.JPG


Puzzle of the Week #245 - Three Digits

I have three digits A B C, such that A is less than B, which is less than C. I can arrange these to form six different 3-digit numbers.

When A is at the start, both possible numbers (ABC and ACB) are prime.

When A is in the middle, both numbers are semi-prime (the product of two prime numbers).

When A is at the end, the number is either abundant by 71 (the sum of its proper divisors is 71 more than the number itself), or deficient by 226 (the sum of its proper divisors is 226 less than the number itself).

What are the three digits?

Puzzle of the Week #244 - Three Way Split

Given a unit square it is possible to split it into three equal areas in a number of different ways. For instance cutting off the top third and then the bottom third. Or cutting off the top third then making a vertical cut to divide the remaining part in two. The total cut length of that first option is 2, whereas the total cut length of the second option is only 1+2/3.

Can you find a way of dividing the square into three equal areas, which requires even less total cut length?

cut sq 3.JPG

Puzzle of the Week #243 - Shaded Circles

A circle of radius 2 has a circular hole of radius 1.

This is done is two alternative ways: in one the hole is tangent to the larger circle forming a crescent shape; in the other the hole is in the exact centre of the larger hole.

In the ‘crescent’ option a shaded circle of radius 1 is drawn, and then infinitely many smaller and smaller shaded circles are added heading off towards the two tips of the crescent.

In the ‘ring’ option just nine circles, each of radius ½ are drawn and shaded.

Which version has the largest combined shaded area?

shaded circles.JPG

Puzzle of the Week #240 - Base 4 Code

I have written a sentence, 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). 

base4.JPG

 

1103 11311 11311 11311 12213210 11311 110211 3233110 3211110230121 1103 1132110311 1103 11311 20331001110 11033 2113333111.

 

Puzzle of the Week #238 - Circle Box

Here’s a fun challenge. The task is to fit circles into a grid according to the following rules:

·         The centre of every circle must lie on a lattice-point of the grid, and no circles can share the same centre point.

·         Every circle must have a whole number as its radius.

·         The circles can touch each other and the edge of the grid, but cannot cross. In other words a circle can be entirely inside another, or entirely outside, but not partially inside.

·         Your ‘score’ is the total area of all the circles you can fit in. A radius-1 circle has area of 1(pi), a radius-2 circle has an area of 4(pi), radius-3 = 9(pi), etc.

·         The example 4 x 4 unit grid has a score of 6(pi) (4+1+1). I have marked the centre of each circle to show that none coincide.

What score can you achieve in the 12 x 12 unit grid?

circle box.JPG

Puzzle of the Week #234 - Split Squad

A junior football team has 15 players, with squad numbers from 1 to 15.

During training they split into two teams: a team of 7 and a team of 8, such that:

The sum of the squad numbers on team A is the same as the sum of the squad numbers on team B.

The sum of the squares of the squad numbers on team A is the same as the sum of the squares of the squad numbers on team B.

The sum of the cubes of the squad numbers on team A is the same as the sum of the cubes of the squad numbers on team B.

How was the squad split into the two teams?

Puzzle of the Week #232 - Five Towns

There are five towns: Anderton, Barmby, Calderwood, Dempsey and Edinburgh.

There are straight roads between Anderton, Barmby and Calderwood.

Dempsey is exactly 25 miles from Anderton, Barmby and Calderwood.

Edinburgh is exactly 12 miles to the closest point on each of the roads between Anderton, Barmby and Calderwood.

How far is Dempsey from Edinburgh?

five towns.JPG


Puzzle of the Week #231 - Mastermind

You may be familiar with the game ‘Mastermind’, where one player forms a code of coloured pegs and another makes guesses. The coder tells the guesser after each guess how many pegs are fully correct, and how many others are present but in the wrong place.

My version is different from the standard version in three respects:

a)    Instead of four, the code is now FIVE pegs long.

b)    Instead of six different colours to choose from, in this version there are only TWO different colours of pegs: RED and YELLOW. There are therefore 2 to the power of 5, (32), different possible codes the coder might have set.

c)    The guesser has decide what their first ‘n-1’ guesses are going to be straight away, before getting any feedback from the coder, such that when the feedback ultimately comes, the guesser can be certain of guessing correctly on the very next guess, the ‘nth’ guess (or earlier if one of the original n-1 guesses happened to have been correct). (In other words, if the guesser decides on 9 guesses and knows that whatever the code happens to be, they can use the feedback from those 9 guesses to be certain that the 10th guess will be correct, then n equals 10).

 

The question is then: what is the minimum value of ‘n’, such that the nth guess is guaranteed to be correct?