Difficult Mathematical Puzzles - 1

Maths number puzzles often feature in a number of competitive exams. These are based on various concepts of algebra, arithmetic, time & distance, etc. Here is a compilation of a variety of puzzles grouped into: easy, medium and difficult maths puzzles. Each article contains a set of 10 mathematical puzzles with answers and explanations. In this article, you will come across difficult puzzles based on the basic concepts of mathematics. 
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Practice the given questions and check your level of preparation:
Q.1.What is the minimum number of numbers needed to form every number from 1 to 8,000?
Example: To form 1122, you would need 2 – 1s & 2 – 2s but you would not count the numbers again that you had already counted from making 1122 to form another numbers.
Q.2.To reach Honululu, in the middle of a barren plataeu, I will have to travel overland by foot from the coast. On a trek like this, each person can only carry enough rations for six days and the farthest we can travel in one day is 25 miles. Also, the city is 125 miles from the starting point. What I am trying to figure out is the fewest number of persons, including myself, that I will need in our group so that I can reach the city, stay overnight, and then return to the coast without running out of supplies. How many persons (including myself) will I need to accomplish this mission?
Q.3. John goes to a furniture auction. All purchases must be paid for in cash. He goes to the ATM and draws out Rs.40,000 (Currency notes are of the denomination of Re.1, Rs.2, Rs.5, Rs.10, Rs.20, Rs.50 and Rs.100). Since John does not want to be seen carrying that much money, he places it in 16 envelopes numbered 1 through 16. Each envelope contains the least number of currency notes possible (i.e. no two tens in place of a twenty).
At the auction he makes a successful bid of Rs.8456 for a table. He hands the auctioneer envelopes number(s) 4, 9, and 14. After opening the envelopes the auctioneer finds exactly the right amount. How many Rs.2 notes did the auctioneer find in the envelopes?
Q.4. An eagle is flying between two trains, each travelling towards each other on the same track at 80 km/h. The eagle reaches one engine, reverses itself immediately, and flies back to the other engine, repeating the process each time. The eagle is flying at 50 km/h. If the eagle flies 150 km before the trains meet, how far apart were the trains initially?
Q.5. Shantanu had three sons, to whom he wants to divide his 19 camels, in the following way: The oldest son was to receive one half the property, the next a quarter, the third one fifth. The three brothers were at a loss as how to divide the inheritance among them-selves without cutting up a camel, until a stranger appeared upon the scene.
Dismounting from his camel, he asked if he might help, for he knew just what to do. The brothers gratefully accepted his offer. Adding his own camel to Shantnu's 19, he divided the 20 as per the will. The oldest son received 10, the next 5, the third 4 . One camel remained: this was his, which he mounted and rode away.
Scratching their heads in amazement, they started calculating. The oldest thought: Are not 10 greater than the half of 19? Someone must have received less than his proper share! But each brother discovered that he had received more than his due. How is it possible?
Q.6. There is a magazine made up of 8 large sheets of paper folded in half. The magazine has 32 pages altogether. The second sheet contains pages 3, 4, 29 and 30. If I pick up a sheet containing page 19, what are the other pages that this sheet contains?
Q.7. How many steps are required by a carpenter to cut a piece of plywood of dimensions a x b into 1 x 1 pieces?
You can cut an existing piece horizontally or vertically.
You cannot cut two or more pieces at once (so no cutting through stacks).
Q.8. Ruchira went out for shopping. She had in her Da Milano handbag approximately Rs. 15/- in one rupee notes and 20 paise coins.When she returned, she had as many one rupee notes as she originally had 20 paise coins. And as many 20 paise coins as she originally had one rupee notes. She actually came back with exactly one-third of what she had started out with. How much did she spend and exactly how much did she have with her when she started out?
Q.9. A thief wants to open a lock with 5 digits as the key to open it. He knows that the 3rd digit is three less than 2nd digit, while 2nd digit is four smaller than 4th digit. The first digit is three times the fifth digit. There are three pairs whose sum is 11 and third and fifth digits are equal. Find the key to open the lock.
Q.10. What is the 504th digit of the series in which natural numbers are written in order i.e., 1234567891011121314…?
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