I had run into this problem long time back. I ran into it again last week. Thought will post it. See if you can solve it without using Google to find the answer. If not, then Google it – pretty interesting, huh? (I will post the answer later).
A basketball player keeps track of his throws for a full calendar year. He, of course, misses a few shots and succeeds with a few more. He missed the very first shot of the year. But he ended the year with 83% successful shots. You have to prove that there had to be a point where is success rate was exactly 75%.
Hint: this is not necessarily true for any number – e.g. 60%. But it is always true for 75%.
In fact, can you guess what are the other % (other than 75%) for which this is also true?
Here is one that will have you think about shapes…
Tomorrow is 4th of July and we will be seeing a lot of flags flying proudly all over the USA. Did you know that there is only one country in the world whose flag is not rectangular in shape?
In any case, imagine a country that has a rectangular flag (See my poor rendering in red) with a smaller rectangle of different color inside it somewhere. (See my blue rectangle). Note that the blue rectangle can be anywhere and in any orientation.
Question is, can you cut the flag to make two identical pieces?
How many of these statements are true?
1. You can see the Great Wall of China from the moon on clear days (my sixth grade teacher had taught us this)
2. Bulls get excited by red color (which explains why they charge the matador waving the red flag)
3. Napolean was rather diminutive (by French standards those days anyways)
4. Only the royals among the Vikings wore the horned helmets
5. Einstein was weak in math (and failed once) as a school kid.
6. We have 5 senses (sight, sound, taste, touch and smell). (I was taught this pretty early in life).
7. Speaking of senses, the tongue has different parts where we taste different tastes (sweet, salt etc etc)
8. Continuing with our body, artistic folks are more active on the right side of the brain and vice versa for the science and math oriented ones.
9. There is no such thing as a “scientific proof”
10. A steep learning curve implies you will have great difficulty learning it.
If you have Googled, which ones surprised you?
Here is a puzzle we were solving this morning.
Each letter stands for a digit. Those digits are 0,1,2,3,4,5
One twist – The letters below the line do NOT match the letters above the line in terms of the digits they represent. In fact, the digit represented by a letter above the line is separated from the digit represented by the same letter below the line by 1. So if C below the line is 2, then C above the line has to be either 1 or 3.
Can you solve the following subtraction? Send me a message with the answer.
Do you know the game Concerto? I did not. Just learnt about it today. The game goes roughly this way –
You start with a grid of squares – can be as big as you want it to be.
Now two players take turns to draw lines. Each turn can draw a line on any of the sides of any of the grid squares – provided somebody had not drawn there already – and it can be of the length of one side at a time only.
(You might remember a variation of this game where anytime somebody completes a grid square, he/she claims that square and in the end you count up who got how many).
However, in this game, anybody who completes a shape – any shape – entirely by his/her own lines only – wins. Note that it does not have to be a square or rectangle – it can be any shape. Also there might be lines drawn inside the shape by any of the players – it does not matter. It just needs to be a completed shape with one player’s lines only.
Look at the picture below as an illustration. The player with lines with black tips wins. Think of the shape comprising the four squares marked with red blobs – that is a complete shape built by the black tip lines only.
Here is the question. Just like the second player in a game of tic tac toe can always prevent the first player from winning, in this game too, the second player can come up with a strategy that will ensure that the first player can never win regardless of how big the grid is.
Can you come up with such a strategy?
(Send me PM with your answers; I will publish your correct answers in the Comments section later)
Remember the puzzle I posted last Saturday? https://www.rajibroy.com/?p=18817 The real problem in the book “The Riddler” (thank you Matt Moore for giving me that book) had 7 in the team – not 3. And the answer the book has 7/8. I am getting a far better probability. What am I doing wrong?
The problem, to remind ourselves is – randomly a black or white hat will be put on each of 7 of us. We cannot see our own color but we can see everybody else’s color. When called upon to guess our own color, we can call Black or White or we can say Pass. If one calls Black or White and is right, then the whole team wins. If it is wrong, then the whole team loses. However, if Pass is called, another person in random is asked to guess the color on their head. If everybody Passes then the whole team loses. What is the strategy to maximize the chance to win and what is that probability?
I am attaching the answer in the book which comes to 7/8 (using XOR logic).
My answer is the following:
The first person : If he sees the same color on the other six, he randomly calls a number. Else he says Pass.
The second person : Now he gets a chance only if the first person called Pass. Which means the first person has effectively said “All of you DO NOT have the same color”. So, if the second person sees the same color on everyone of the rest five, he simply calls the opposite color. He is guaranteed to be right. If he sees NOT all hats with the rest five to be of the same color, he simply says “Pass”.
The third person : uses the same logic. If the rest four have the same color, simply call the opposite color. Else say Pass.
This will go recursively and somebody is guaranteed to get the right answer. (In the absolute extreme case, The fifth person will call Pass – which is telling the sixth and seventh person – “Hey you two have different colors” and the sixth person can see the seventh person’s hat color….)
So independent of how many original players were there, there are only 2 cases the team loses – when the first person saw everybody else having the same color (all white or all black) and his random guess of his own color turned out to be wrong.
So, they lose with a probability of 2 / (2 the power n)
Winning probability is (1 minus the above) – which is much higher than 7/8
What am I doing wrong?
This is adapted from a more complicated version submitted by Jared Bronski in “The Riddler”. (Thanks to Matt Mooore who gifted me the book this week).
It is one more of those hats on your head and strategy formulation question.
There are three of you. Randomly a white or black hat is put on your head. Each one of you can see the color of the hat on the other two but not the one that is on your own head.
One by one, you will be asked to guess what color hat you have on your head.
Your options are (*) tell a color – Black or White or say (*) Pass.
At any point, if somebody calls a wrong color – you all lose. But if somebody calls a right color – you all win. And if everybody passes at the end of first round, then you all lose.
What strategy can you formulate beforehand to maximize the chances of winning? And what is that probability?
Read this up in the book “Mathematical Circus”.
You have 2 green balls, 2 yellow balls and 2 red balls. One ball of each color is 11 pounds each. The other ball of each color is 9 pounds each. You have a scale and pan balance. (meaning you can compare the weights of two sides – which is heavier and which is lighter but you never know the exact weight).
What is the least number of weighings required to find out which are the three heavier balls and which are the lighter balls?
Send me personal message with the reasoning.
I have finally been able to memorize all the country names and their capitals. For this purpose, I am defining countries as those 195 that are recognized by the United Nations. Some interesting puzzle questions emanate from that:
Take a guess at these questions and write down if you want in the Comment section. Then check with Google (or wait for a day – I will publish the answers). Do NOT write the answers here AFTER Googling.
1. The smallest number of letters in any country capital is 4. For example: Rome. How many such 4 letter country capitals are there? Can you name them?
2. How many countries have the capital name same as the country name? e.g. Singapore capital is Singapore. How many can you name?
3. Now, some capitals are the same as the country name with the word “City” attached. e.g. capital of Mexico is Mexico City. Similarly, what are the other countries that have the same pattern of capital names?
4. Many countries, during their early ages, had certain cities flourish because of trade – that happened mostly thru sea waters. Thus you will see many of those countries have a port as their capital. Some of them even have the name Port in their capital name. e.g. “Port Au Prince” for Haiti. What are the other capital names that you can come up with that have Port in it?
5. Many capital cities were named after certain Saints. They tend to have names starting with St., San, and so on. How many of them can you come up with?
6. Which are the capital cities that start with the letter “Y”?
7. How about “Z”?
8. Which is the most common first letter for capital names? Meaning more capital names start with this letter than any other letter. There are an astounding 25 of them!
9. A couple of countries have an apostrophe in their capital names. Can you name them?
10. Now the final one: What is the capital (legislative capitals) of Sri Lanka and Myanmar? Hint: I did not realize that they had moved their capitals recently.