Maths Puzzles

A scale is balanced. On one side of the scale, there are 3 1 ⁄ 2 bags of rice and on the other side there are 1 3 ⁄ 4 bags of rice and 1 ⁄ 4 kilo weight. What is the weight of one bag of rice? This puzzle is borrowed from the book "Brain teasers" by Ian Livingstone and Jamie Thomson Show answer Let the weight of 1 bag of rice be b. So 3 1 ⁄ 2 b = 1 3 ⁄ 4 b + 1 ⁄ 4 Transferring  1 3 ⁄ 4 b to LHS we get 1 3 ⁄ 4 b =   1 ⁄ 4   b =  1 ⁄ 4  /  1 3 ⁄ 4 =    1 ⁄ 7 kg

  Can you add two to eleven and get one as the answer? How?   Show answer Answer : When you add 2 hours to 11 o'clock, you get one o'clock.    The question is borrowed from the book "Brain teasers" by Ian Livingstone and Jamie Thomson.

A bag contains some red pencils and some blue pencils. The ratio of red to blue pencils is 7:11. If the total number of pencils the bag is 90, what is the number of blue pencils? (This question is borrowed from the book "Competitive Maths for middle school" by Vinod Krishnamoorthy)  Show answer Answer : 55 pencils   If the total number of pencils is 7+11 = 18, then 11 is the number of blue pencils. So the ratio of blue to total is 11:18   Since the total number of pencils is 90, number of blue pencils is 11/18*90 = 55 pencils

There are 5 consecutive even numbers whose sum is 640. Which are these numbers? e.g. 2,4,6,8,10 are consecutive even numbers and their sum is 30. Show answer The average of 5 consecutive even numbers will be equal to the third number. Because if the numbers are  x,x+2,x+4,x+6,x+8, then their sum is 5x+2+4+6+8 = 5x+20  Their average is (5x+20)/5 = x+4 which is the third number of the sequence.   Here the sum is 640 => average is 640/5 = 128   Hence the sequence is 124,126,128,130,132 

An explorer Dr. Franz was trekking across a desert in search of a secret treasure. But the water container in his jeep was leaky.  In the first hour, the container lost half of the water plus one litre. In the second hour it again lost half of the remaining water plus one litre. In the third and fourth hour same thing repeated. But by the end of fourth hour, the container was empty. How much water did the container originally hold?   Show answer Answer = 30 liters   Let us say there was x liters of water to start with.  First hour he lost x/2 liters +1  First hour remaining water was x - (x/2+1) = x/2 - 1   Second hour he lost (x/2-1)/2  + 1 = x/4 -1/2+1 = x/4 +1/2 Second hour remaining water was x/2 -1 - (x/4 +1/2) = x/4 - 3/2   Third hour he lost( x/4 - 3/2 )/2 +1 = x/8 +1/4 Third hour, the remaining water was x/4 - 3/2 - x/8-1/4 = x/8 - 7/4   Fourth hour he lost (x/8 - 7/4)/2 +1 = x/16- 7/8 + 1 = x/16 + 1/8 Fourth hour, the remaining ...

 Answer is 35 Let us use first letters of their names as number of drones shot by each pilot. I = 10+J ------(1) L = I - 4 ------(2)    => L = 6+J J + V = 9 ------(3) L = 3+V------(4).   =>6+J =3+V => V - J = 3 V  =  C + 2 ------(5)    V+J = 9 V-J = 3  Adding these two, we get 2V = 12   => V = 6.  and J = 3   L = 6+J = 9 I = 10+J = 13 C = V - 2 = 4 Total number drones shot down is sum of all these= I+J+L+V+C = 13+3+9+6+4=35     

This puzzle is from the book "Brain teasers" by Ian Livingstone and Jamie Thomson. Earth was under attack by marauding Mechanoids—robotic machine aliens from  another world. Five Earth Force pilots returned from a sortie into space after destroying several Mechanoid Battle Drones.    Ibrahim shot down 10 more than  Juan. Luke shot down 4 less than Ibrahim. Juan and Vladimir shot down 9  between them. Vladimir shot down 2 more than Chi, but Luke shot down 3 more  than Vladimir.   How many Battle Drones were shot down in all?     You can find the solution here .  

 It is not. Because your wife is widow, only if you are dead. A dead man can't marry. The man is living in the ground floor of a high rise building. So even when he jumps out of his window, he has not fallen down from a height and is perfectly fine. The man has entered a lift and has pressed a button to go down. When the lift is going down, his weight is temporarily reduced due to the downward motion of the lift. And when the lift stops and man gets out of the lift, he has his original weight.  

Is it legal in India for a man to marry his widow's sister? Justify your answer. A man living in a high rise building decides one day to go out of the window instead of door. He jumps from the window. But he survives the fall without a scratch, and continues to walk to his office. How is this possible? A man enters a room and presses a button. Immediately he is 10 kg lighter. However when he leaves the room, he is having his original weight. How?      Find the solutions here . 

 You break all the four pills into halves. And you take one half of each pill and give the other half to the guide. Now both of you have eaten 1/2 blue pill+ 1/2 green pill + 1/2 blue pill+ 1/2 green pill i.e. 1 blue pill and 1 green pill.  And the antidote will work. 

  You decide to go on a journey to Amazon forests. Before starting the journey, you go to the pharmacist and buy antidote for poisons. The pharmacist gives you 4 pills, two green and two blue. He warns you that you must take one blue pill and one green pill in order for the antidote to work. Taking two green pills or taking two blue pills will not work. Later during the trip, one day as you and your guide are walking, a snake falls down from the tree and bites you. The guide tries to help you, but the snake bites him too.   He informs you that the snake is highly poisonous. But you tell him not to worry as you have antidote for poison. You open your bag. To your dismay, the caps of both pill bottles have opened and the bottles are empty. All the four bottles are at the bottom of your bag.  But you can't tell the colors of the pills as it is already quite dark. How do you save yourself and your guide? Solution : In this post  

They decide that everyone will say white if they see odd number of white hats in front of them. If they see even number of white hats in front of them, they say black. Let us see how this works. The last person sees ahead and let us say he sees odd number of white hats and he says "white" The person in front of him sees ahead. And if he sees odd number of white hats in front of him, he realizes his hat must be black and he says "black". Instead if he sees even number of white hats in front of him, and since the person behind him had seen odd number of white hats, he realizes his hat is white and says "white". The third person also repeats the same procedure.  This is repeated until you reach the last person. If the person behind him says 'white', his hat is white and he says 'white'. If not he says 'black'. Let us see another possibility The last person sees even number of white hats and says 'black' If the second person see...

10 prisoners share a cell. The prison warden tells them that the next day they’ll be randomly lined up for execution. He’ll place either a black or white cap on everyone’s head. Each prisoner starting from the back of the line will guess what color his hat is. If he gets it wrong he’ll be executed.  Every prisoner can not see the hat on his head or of those behind him. He can see the hats of everyone in front of him.   How can the prisoners devise a strategy to save their lives?    You can find the solution here . 

Since there are 10 people in the party, and since a person can not shake his own hand or that of the spouse, the number of handshakes could have been 0 to 8.  Since he got different answers from 9 different people, the answers he got must be 0,1,2,3,4,5,6,7,8 . Now a person has shook 0 hands. Which means the person who shook 8 hands must be his/her spouse. Because he has shook hands with everyone except his/her spouse.  0,8 is one couple. Next there is another person who has shook only one hand. He must have shook the hands of 8-shaker person. Which means his spouse must have shook 9-spouse-zero shaker = 7 hands. 7,1 is another couple.  There is 2 hands shaker. He has shook hands with 8 shaker and 7 shaker. His spouse must have shook 6 hands. If we continue like this, we get (0,8), (1,7),(2,6),(3,5),(4,4) But since all answers were unique, only one person told he/she has shook 4 hands. Which means the other person who shook 4 hands is you. Which means your spouse has shoo...

 This one is quite an interesting puzzle. I took the help of chatGPT to solve. Even then it took a while to understand the solution. You are at a dinner party with 5 couples. That evening, no person shakes hands with their partner. At the end of the party, you ask all 9 other people how many different hands each shook, and get 9 different responses. How many hands did your spouse shake?   You can find the solution here.    

 He needs 58 seconds.  When he cuts the the 29 th   pieces, the last piece is already cut. He must make only 29 cuts. 

 A jeweler has a golden wire of 30 cm. which he should cut into 30 pieces of 1 cm each. If he takes two seconds for each cut, and he cuts all of them without break, how much time does he need?   Find the solution here .  

 When the dealer gives you a discount of 15%,  amount will be x (1-0.15) = 0.85x Then when he takes a commission of 10%, his commission will be 0.085x The final amount you will pay  is 0.85x + 0.085x = 0.935x  When the dealer first takes a commission of 10%, amount will be 1.1x When he gives a discount, discount amount will be 1.1*0.15x = 0.165x The final amount you pay will be 1.1x - 0.165x =  0.935x For you, both the options are equal.   For the dealer -   If he takes a commission after discount, his commission will be 0.085x = 8.5%  If he takes commission before discount, his commission will be 10% So for the dealer second option is more profitable.    

 You want to buy a car. The dealer gives you two options. He gives you a discount of 15% and then charges a commission of 10% He first charges a commission of 10%, then gives a discount of 15% Which option is more profitable to you?  Which option is better for the dealer?  You can find the solution here .  

 The traveler cuts the third link. On the first day, he pays this cut link. Second day, he pays two links and gets back one cut link.  Third day, he pays three links and gets back two links. Fourth day, he pays cut link again. Fifth day, he pays two links and gets back cut link. Sixth day, he pays the cut link again.   

 A traveler has no money, but has a gold chain with 6 links. So he decides to pay one link per day of his stay in the hotel.  The hotelier agrees with the condition that he will accept only one cut link.  How does the traveler stayed for 6 days and paid for all of these days.  Find the solution here .    

 This puzzle is quite simple. Let us say John has worked for x days. Which means he earns x dollars each day. So total money he gets is x 2   Robert earns one dollar more. Robert earns x+1 dollar per day. He worked one day less. The number of days he has worked is x-1.  Robert earns (x+1)(x-1) =  x 2  -1 dollars John earns more.  

John earns same amount each day which is equal to the number of days he has worked.  Robert earns one dollar more than John, but works one day less. Who has more money?    Find the solution here .  

 Since the lines oj and ol are radii, they are both equal to r. Which means the triangle is an isosceles triangle or equilateral triangle. Now since the sides oj and ol are equal, their opposite angles ojl and olj are equal. So each of these angles are (180 - 60)/2 = 60. Which means the triangle ojl is an equilateral triangle. Which means its area is  √3 / 4 * r 2      Now area of sector jol is = πr 2 * 60/360 =   πr 2 /6

 There is a triangle Ojl whose two vertices j and l are points on a circle and third vertex O  is the center of the circle.  If the angle jOl is 60, what is the area of the triangle? What is the area of the sector Ojl?  By the way, this puzzle and its solution was given to me by chatGPT.   You can find the solution here .  

Solution : 0 or 1000  Let us look at the possible combinations.  FFT FTF TFF  If only Harry is right, David owns at least one book. But if he owns more than 1000 books, Harry is right. Which means FFT is not a possible solution. If David owns more than 1000 books, if Thomas is right, then he certainly has at least one book, which means if Thomas is right, Harry should also be right. Which means TFF is not a possible solution. If David has no books at all, then Thomas is wrong. And Harry is also wrong. But Richard is right because fewer than 1000 can also be zero.  The other solution is David has exactly 1000 books. If he has 1000 books, Thomas is wrong, Richard is also wrong, but Harry is right - as David has at least one book.    

 Three statements are made about the books owned by David. Thomas - David owns more than thousand books.    Richard : No, he owns fewer than that.   Harry : He certainly owns at least one book.  Only one of these three statements are true. Who is right?    Find the solution here .  

 Answer : 10 cm   Solution: In fact the solution is quite simple.  Let us look back at the diagram. DC is 5 and also c to the next point on circle is also 5 as is shown in the diagram. As D is the center of the circle, the radius is 10 units. As B is also a point on the circle, DB is radius and hence DB = 10. DB is the diagonal of the rectangle. So the other diagonal AC is also 10cm.  

 A rectangle is inscribed in a quadrant of a circle with the distances as shown in the diagram. Find out the length of diagonal AC.  Hint : the solution is much simpler than you think.     You can find the solution here .  

Answer : 4 units  Area of a hexagon which is inscribed in a circle of radius r is 3 √3 / 2 * r 2  This hexagon has its side = radius of circle.  But the circumscribed hexagon has radius of the circle as its perpendicular to a side from center of circle.      As shown in the diagram, if the side of outer hexagon is s, then the radius from center of circle perpendicular to the side, cuts the side in half. So the radius forms a right angled triangle with three sides as r, s/2 and hypotenuse as s. Using Pythagoras theorem s 2  = r 2   +s 2  /4 3s 2  /4 = r 2  s 2   =r 2  * 4/3 Now replacing this value for the area of outer hexagon, we get area =  3 √3 / 2 *s 2   =  3 √3 / 2* r 2  * 4/3 But area of inner hexagon is 3 =>    3 √3 / 2* r 2     is 3 Area of circumscribed hexagon = 3 * 4/3 = 4 units. ...

2 Regular hexagons are inscribed and circumscribed in a circle. If the area of inscribed hexagon is 3 units, what is the area of circumscribed hexagon?   Find the solution here .    

Answer : 96 cm   The original width of 9cm. After 3 invocations of magic spell the width of the belt becomes 9/3 , 3/3, 1/3  = 1/3cm The final area is 4 Therefore final length is 4/width = 4/1/3 = 12 cm. The length before 3rd invocation is 12*2 = 24 cm Length of the belt before 2nd invocation is 24*2 = 48 cm. Length of the belt before 1st invocation is 48*2 = 96 cm.     

There is a magic belt. When ever the wizard utters a spell its length reduces to half its original length and width reduces to 1/3 of its original width.  The original width was 9 cm. After the wizard invoked the spell 3 times, the area of the belt now is 4 cm 2  . What was the original length of the belt?    Find the solution here .