The Babuschka and the pearls

Before going home, Inspector Simon Mart visited a pearl shop on the island of KoaLoao, where every native was either a TruthTeller or a LieSpeaker, he decided he really had to take home a souvenir.

He looked around in one of the local pearl shops. It was not very large, but his eye fell on a nicely crafted Babuschka – one of those russian dolls where, when you open it, it contains another similar doll, which, when you open it, ok, you get it.
This babuschka contained 3 smaller dolls, and had six small pearls inlaid for the eyes of the dolls. He picked it up and weanted to buy it, but his eye fell on a pillow with 4 beautiful large pearls, some entire white, some entirely black. He remebered that all pearls on this island were either black or white.
“How many of these large pearls do you have?” Simon asked, interested.
“Not very many” the shopowner asnwered. “My neighbour next door has more, and he has 29 pearls of this size.”
“That doesnt tell me how many you have” the inspector remarked.
“Well, if I would put all my large pearls in a bag, both black and white, and you would take out two at random, the chances would be exactly 1 in 5 that you would have two black ones.”
“Don’t believe a word he says!” the servant in the shop warned him. “My boss is a notorious LieSpeaker! Our neighbour has 30 pearls of this size, and the chances are exactly 1 in 4 that you would take out 2 white pearls!”

“That makes things clear!” answered inspector Mart. “Thank you!”

How many pearls does the shop owner have? And who is speaking the truth?

Please try to solve the puzzles on your own. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

When you have solved this puzzle, you can check your solution here

Morozkin problem

Vladimir Arnold in the April 1997 edition of the Notices tells:

The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

This problem can be found at several places on the web, and I assume there is no harm in reproducing it here.
I would like to encourage you to solve the puzzles on your own. It will increase your self confidence, while looking up the answer will lower your self esteem.

When you have solved this puzzle, you can check your solution here

You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Divide the clock

The illustration shows an old fashioned analog clock.
Usings two straight lines, in how many parts can you divide it so that the digits on all parts have an identical sum?

I found this puzzle on aplusclick.com as a former math olympiad problem.

I would like to encourage you to solve this puzzle on your own. It will increase your self confidence, while looking up the answer will lower your self esteenm.

When you have solved this puzzle, you can check your solution here

You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the solutions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Letterboggle

This puzzle type has nothing to do with the game “letter boggle”.

As usual, the rules are simple:
– fill in the entire alphabet, using every letter exactly once;
– consecutive letters are adjacent horizontally, vertically or diagonally;
– a letter in the border means that the letter appears in the indicated diagonal, row or column;

In the following puzzle, some letters are given to help you start:
1) Letterboggle^*

I found this type of puzzle in “PUZZEL”, a year-end addendum to the newspaper reformatorisch dagblad. It had 2 puzzles of this type, copyrighted M. Balmaekers, though Mr. Balmaekers used a 5×5 square and dismissed the letter Q. I intend to have more puzzles like this in one of the upcoming e-books.

I would like to encourage you to solve this puzzle on your own. It will increase your self confidence, while looking up the answer will lower your self esteenm.>When you have solved this puzzle, you can check your solution here

The stairs in the tower of Sheik Oil Well

Did you ever wonder why medieval sultans have such beautiful daughters? Or why they have such complicated ways to select their suns in law?

1) Sheik Oil Well^*
Sheik Oil Well had two huge towers in the middle of the desert. His oldest daughter was renowned for her beauty, and suitors for her fair hand came from all the neighbouring oases.
The sheik told them: Do you see the left tower? He who can tell me how many steps the stairs have, without going up and counting them, will win her hand. I will tell you this: the number of stairs is three more than a prime number. When you would run up with 5 steps at a time, you would find that your last jump would only be 4 steps. When you divide the number of steps by 7, you would have 5 left. Oh, and the number of steps is a multiple of 19.

What is the smallest possible number of steps?

You can check your solution at here

I wish you everything that is good for the new year!

Please try to solve the puzzles on your own. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.