The numbers 1 – 9 and combinatorics.

As children, we all learned to count. Thus we rarely think about counting as ‘difficult’. Yet mathematicians have developed a special branch of mathematics for the art of counting. The branch is called Combinatorics. Typical questions in Combinatorics are:
1) In how many ways can a stack of 52 playing cards be arranged?
2) When we have a vase with 5 black and 5 white balls, in how many sequences can we pull them out?

In today’s problems, we work with the cards 1 to 9:

1) How many ways?^*/*****
It is easy to arrange these cards into 3 groups, all with the same sum:

One reason it is easy, is because there are several solutions. Withe sum of the 9 cards being 45, each of the three groups will have to have sum 15. But in how many ways exactly can we divide the cards 1-9 into three groups, all with the same sum?

2) Be creative^****/*****
Now be creative in the arrangement of your cards. In how many ways can you create 3 groups in such a way that the three groups still all have the same sum, but the sum is not 15?
Yeah, you may cheat in this problem. But your cheating is limited to arranging the cards.

3) Combinatorics (unsolved)^*****/*****
The sum of the first n cards is n(n+1)/2. To divide these number into three groups with the same sum, either n or n+1 mus be a multiple of 3. So this is not possible for n=4, 7, 10 and so on.
Here is a short list
n
2: 0
3: 0
5: 1 (5, 1-4, 2-3)
6: 1 (1-6, 2-5, 3-4)
8: 4
9: see answer to problem 1.
Now can you find a general formula for the number of possible groups?
Or for a simpler start: in how many ways can we draw cards from a series 1-n in such a way that that the sum is some given number?
Or: can you construct an algorithm that shows that the cards 1-3n (n>=2) can always be divided into three groups with the same sum?
I don’t have the answers to these questions, they just look interesting to me.

A new puzzle is published on Fridays, at least twice a month. You may check your solutions here.

The successor of the sultan (3)

The ways in which the sultan of Buhrundipur choose his successor, were a bit, uh, unusual. He presented them with a series of tests. One by one he lead the into an empty basin with four doors.

Test 3^***/*****

“Only one door hides a treasure, the other doors hide a hungry shark. Oh, and only one label is true. Good luck!

Which box should the candidate open?

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solution here.

Complete this multiplication

1) Complete this multiplication^**/*****
Fill in the missing numbers such that
54 x 2__ = ____8
is a correct multiplication and all the digits 0 – 9 are used exactly once.

This problem was published in the Dutch mathematics magazine Pythagoras, issue no 4 in year 9 (1970).

New puzzles are published at least twice a month on Friday.
You can find the solution here.

Fibonacci series

Today’s puzzles were inspired by Michael Jacobs “Introductory Thinking Tasks”.

You all know the Fibonacci series: a term is the sum of the two previous terms, with the first two given. The top row shows a classic sequence:

Your task is of course to fill in the empty spaces of the other two rows.

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solution here.

The successor of the Sultan (2)

The sultan took the survivors of the first round one by one to a second room with three boxes, depicted below. One was made of bronze, one of silver, and the third of gold.

Test 2^***/*****
He told them: One of the three chests holds a treasure. And exactly one of the labels on the boxes is true. You must open one of the boxes. No doubt it will comfort you that the poison of the snakes in the other two boxes are Naja mortiferum, the toxin of which is extremely painful and deadly, but very fast working, so you won’t suffer very long.

Which box should the candidate open?

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solution here.

The sum of the digits

If you think that mathematics and arithmetic is something for boys, you probably never read about Shakuntala Devi. Born in 1929, as a young child she was taken by her father on road shows to display her abilities for mental calculation. At the age of six she demonstrated her abilities at the University of Mysore.

In 1977, when she must have been 48, at Southern Methodist University, she gave the 23rd root of a 201-digit number in 50 seconds.[6][4] Her answer, which was 546,372,891, was confirmed by calculations done at the US Bureau of Standards by the UNIVAC 1101 computer, for which a special program had to be written to perform such a large calculation, which took a longer time than for her to do the same.

She wrote several books, on puzzles, astrology, memory and homosexuality. What interests us here is the book “Puzzles to puzzle you”.
Here is one of them:
Which number is exactly 3 times the sum of its digits?

In her honor :
Which number is exactly 11 times the sum of its digits?

You can check your solution here

The successor of the Sultan (1)

It was already known that the Sultan of Buhrundipur was a cruel men. It is only through recent research that we learnt how cruel. We already knew that the empire collapsed after his demise. He put all candidates whom he considered to be a possible successor through a series of tests, which have only now come to light.

Test 1^**/*****
There are 3 boxes, each with a label. One box holds a treasure, two boxes hold a deadly poison which is released when opening the box. One label tells the truth, the other two are lies.
Box 1: “This box holds a poison”
Box 2: “This box holds a treasure”
Box 3: “The label on box 2 is false”

Which box should the candidate open?

New puzzles are published at least once a month on Fridays. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solution here.

Koprol

This Friday we pose a word problem. A quote from Dr. Martin Luther King has been hidden in this picture. Start at the blue circle, and move horizontal or vertical until you have used all letters. The last symbol is the dot. The green squares represent blanks. No diagonal moves are allowed.
I found this type of puzzle in the Dutch newspaper Algemeen Dagblad

New puzzles are published at least twice a month on Friday. You can check your solution here.

Alphametics – double true – English

An Alphamatic is a puzzle in which words are part of a calculation. An example is
SEND
MORE
—– +
MONEY
which, if the letters are replaced by the correct digits, forms a correct addition.

A special class are the “double true” alphametics. THese are the puzzles I want to have a look at in this post.
1) I listed one before:
THREE
FIVE
FIVE
SEVEN
TEN
——+
THIRTY

The university of Bieleveldt has a nice list:
2) SEVEN + SEVEN + SIX = TWENTY difficulty: 2
3) EIGHT + EIGHT + TWO + ONE + ONE = TWENTY difficulty: 3
4) ELEVEN + NINE + FIVE + FIVE = THIRTY
5) NINE + SEVEN + SEVEN + SEVEN = THIRTY
6) TEN + SEVEN + SEVEN + SEVEN + FOUR + FOUR + ONE = FORTY
7) FOURTEEN + TEN + TEN + SEVEN = FORTYONE
8) NINETEEN + THIRTEEN + THREE + TWO + TWO + ONE + ONE + ONE = FORTYTWO
9) FORTY + TEN + TEN = SIXTY

This list is, however, not exhaustive. Here are a couple more:
10) THREE + THREE + TWO + TWO + ONE = ELEVEN difficulty: 3
11) ELEVEN + THREE + THREE + THREE = TWENTY difficulty: 4
12) ELEVEN + THREE + THREE + ONE + ONE + ONE = TWENTY difficulty: 4
13) SEVEN + THREE + THREE + TWO + TWO + ONE + ONE + ONE = TWENTY difficulty: 4
14) SEVEN + FIVE + TWO + TWO + ONE + ONE + ONE + ONE = TWENTY difficulty: 3
15) SEVEN + FIVE + TWO + TWO + TWO + ONE + ONE = TWENTY difficulty: 3

I feel there are a couple more lurking out there, but I hope to list more of them in a future post.

They exist in many languages, about a year ago I published some Dutch ones. 

New puzzles are published at least once a month on Fridays. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.

Four equal sized areas

This week we have some calculation problems for you.

1) four areas 4×6^**/*****
Divide this grid into four equal sized areas, each with the same sum.

2) four areas 4×7^**/*****
Divide this grid into four equal sized areas, each with the same sum.

You can check your solutions here.

New puzzles are posted twice a month on Friday. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars. You can check your solutions here.