4. Numbers Don’t Lie

“A good decision is based on knowledge, and not on numbers” – Plato

Some of us love Maths and some hate it. Love or hate, we cannot deny that numbers are integral to our lives. We have to deal with numbers on a daily basis, and a number of times our proficiency in Numerical ability determines whether we win or lose in a situation.

Numerical ability does not really need a degree in Maths, nor does it involve complex mathematical formulae and concepts. It is something that even a middle school student can possess in adequate measure. You just need to set aside your fear of Maths and learn to trust your ability to do basic mathematical calculations.

Most of the puzzles which follow won’t even require putting pen to paper, and those that do will involve simple calculations mostly. What these puzzles will test, however, is the ability to grasp the basic concepts and to figure out exactly what you must do to arrive at the answer.

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4.1    This is a question from the Oct’ 80 SAT test, where there was supposed to be only one answer, but the people who prepared the test overlooked something which meant that the choices given contained two correct answers to this question. When this error came to light, those who had marked either of the answers were given full marks for this.

Can you find what the mistake in the below question-answer set was?

Which row contains both the square of an integer and the cube of a different integer?

(A) 7, 2, 5, 4, 6

(B) 3, 8, 6, 9, 7

(C) 5, 4, 3, 8, 2

(D) 9, 5, 7, 3, 6

(E) 5, 6, 3, 7, 4

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4.2    Complete the given number sequence:

1,2,4,7,11, ?, ?

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4.3    Think you are fast in doing calculations mentally? Lets see how quickly you can figure this one out.

The total cost of a bat and a ball comes to Rs. 230. The cost of the bat is Rs. 200 more than the ball. What is the cost of the ball?

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4.4    While you may choose to do this one through trial and error, but I recommend that you use logical reasoning or, alternatively, write it in the form of a couple of simple equations and solve.

Hint: Recall the basic divisibility rule for 9

Which is the smallest two digit whole number which is equal to nine times the sum of its digits?

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4.5    This puzzle is a classic one which forces us to suspend and overcome our initial disbelief in the premise of half a hen and half an egg before we can reach the conclusion.

A hen and a half lay an egg and a half in a day and a half. Can you tell how many eggs will half a dozen hens will lay in half a dozen days?

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4.6    The Binary notation challenges the way we are used to looking at numbers.  Breaking the shackles of conventional way of thinking also is one of the bigger challenges we often face when trying to figure out creative solutions to problems or when trying to innovate. Let’s give it a whirl now.

Take a look at the Binary notation for the decimal numbers 1 to 8 given below:

1=0001, 2=0010, 3=0011,4 =0100,5=0101, 6=0110, 7=0111, 8=1000

Can you say how the number 12 should be represented?

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4.7    On my way to the Himalayan abode of Abujhgarh, I met a man with 7 wives, each of whom carried 7 sacks. Each Sack had 7 hen and each hen had 7 chicks. Man, wives, sacks, hen and chicks, that’s quite a jamboree.

Can you tell me total how many heads were going to Abujhgarh? 

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4.8    Simple number puzzles expressed in sentences often leave us stumped. Once again, we get so used to seeing numbers as symbols and digits only, that many have a hard time relating the sentences to their numerical form. Such puzzles build up our powers of mental association and help in boosting our creative visualization processes.

There is a number from which if I subtract twice the cube of it then I end up with the square of it.

What is the number?

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4.9     Here is another simple puzzle that quite a few people find hard to comprehend, as they lose track of the reasoning process midway.

If 10 people eat 10 Burgers in 10 minutes, how long will it take for 35 people to eat 70 burgers?

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4.10 Just like magicians relying on distraction and sleight of hand to fool the audience into believing something that is not, there are some classic puzzles that rely on getting your thoughts so jumbled up using a clever play on words that you tend to lose track of facts in front of you. Give this one a try, and see how you fare:

Three friends went out to dinner at their favorite restaurant. There was a new steward who took their order. After the meal, the steward brought a bill for Rs 150, so the three friends contributed a Rs 50 note each. The Manager, who was just coming out of his office, recognized the three as old friends. When the steward returned with the payment, the Manager instructed him to return Rs 50 to them.

The steward was dishonest and returned only Rs 10/- to each friend, pocketing the remaining Rs 20.

So now, each friend has paid Rs 40, which comes to a total payment of Rs 120/-.  The steward has kept 20/-. This brings the total to Rs. 140.

Where did Rs. 10 disappear?

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4.11 Number sequences are a wonderful tool to build up our analytical reasoning skills. They encourage us to explore myriad ways a sequence of numbers may be related, and to distinguish patterns out of seemingly disparate elements.

What number should replace the question mark here?

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4.12 If the previous one was too easy, then have a go at this one:

There are two possible answers, can you get both?

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4.13 What numbers should come in place of the question marks?

18,26, 23, 23, 28, 20, ? , ?

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4.14 Making sense of a jumbled data is something we have to all the time at the workplace. Such puzzles improve our skill at organizing data. Be aware, though, that sometimes you really need to think beyond the usual in order to get to the solution.

Create an equation with only the elements given below. Each element must be used only once:

2, 3, 4, 5, =, +

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4.15 At times our concentration on performing calculations and getting to the result is so total that we fail to understand the problem itself.  Can you calculate the correct answer to this one?

A hungry monkey jumped down an almost dry well with slippery sides to get to some coconuts at the 30 feet deep base. Now it must climb out, but it is hard work. It can climb up 7 feet during the daytime every day, but then slides down by 6 feet every night while it tries to rest.

How many days will it take to climb out of the well?

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4.16 This is a puzzle that went viral a few years ago and has been immensely popular since then. There are two answers possible here, so even if you know one then try to work out the other!

1+4   =  5

2+5   = 12

3+6   = 21

8+11 =  ?

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SOLUTIONS TO CHAPTER 4

4.1      While the answer to this was only supposed to be (B) which contains the square of 3 and cube of 2, the paper-setters missed out on the fact that in option (C) there was the square of -2 (square of a negative integer is positive) as well a s the cube of 2, and both are distinct integers !

4.2      This is an easy one, with each successive number increasing by 1,2,3,4,5… and so on.  Thus, the next two numbers in the series will be 11+5=16 and 16+6=22

4.3      If you came up with Rs. 30 as the answer, then look carefully at the wording of the problem. If the ball is Rs. 30, and the bat is Rs. 200 more than the ball, then the cost of the bat itself will be Rs. 230, and hence the total will become Rs.260.

The correct answer is Rs. 15, which means that the bat is for Rs. 215.

4.4      This can be solved in two ways. One is by solving a simple equation. Since it is a two-digit number, the number can be represented as (10x+y), where x and y are the two digits.

Now, since the number is 9 times the sum of the digits the equation can be represented as (10x+y)=9(x+y), which implies that x=8y.

The only single digits that meet this condition are x=8 and y=1

Hence the number is 81.

The second way is by logical deduction. One basic property of any two digit multiple (actually, this is true for all multiples of 9) of 9 is that the sum of the digits always adds up to 9.

Since the number is 9 times the sum of the digits, the answer is 9*9=81  

4.5      1.5 hen =1.5 egg in 1.5 days. Half a dozen days is 1.5*4=6

Therefore 1.5 hen will lay 4 times as many eggs in 6 days.

Thus, 1.5 hen will lay 6 eggs in 6 days

Hence, 6 hen will lay 6*4= 24 eggs in 6 days.

4.6      Following the same sequence, 9 will be represented by 1001, 10 = 1010, 11=1011. Hence, 12 will be represented by 1100.

There is a methodology to work out how any number may be represented by its Binary equivalent. It goes like this:

  1. First  you determine whether the given number is even or odd. If it is even, then assign the digit 0 and if the number is odd, then assign the digit 1. This becomes the first digit from the right.
  2. Now, deduct this last digit so obtained from the original number, and divide the resultant by 2. Again see if the number is even or odd, and again assign the digit 0or 1 accordingly. This gives the second digit from the right.
  3. Repeat the above process till you finally get 1 or 0 as the resultant, when the process comes to a stop.

e.g.: Let’s represent 21 as a Binary: 

21 is odd, so the right-most digit is 1.

(21-1)/2=10, which is even, so the next digit is 0.

 Now (10-0)2=5, which is odd, therefore the third digit is 1.

Then (5-1)/2=2, so next digit is 0.

Finally, (2-0)/2=1.

Thus, 21 is represented by the Binary 10101

4.7      This one is a variation of the classic puzzle known as ’As I was going to St. Ives”, which began as a nursery rhyme in the seventeenth century, The puzzle was made famous by appearing in the movie “Die hard with a vengeance”, where the villain threatens to set off a powerful bomb unless Bruce Willis and Samuel Jackson can solve this is 30 seconds. Eventually they save the day with a second to spare. How did you fare on this one?

The answer is: Only one head is going to Abujhgarh: Me. I only met the group on my way up.

The puzzle tries to mislead you through all the 7s thrown in, which tricks you into trying to do multiple calculations.

4.8       The number is -1.

The cube of -1 remains -1, while the square becomes positive 1.

Hence as per the problem statement, -1 – 2*(-1) = -1+2 = 1

4.9      Since ten people are eating 10 burgers in 10 minutes, it means each person has eaten one burger individually in 10 minutes.

Hence, 35 people will eat 35 burgers in 10 minutes only.

Therefore, 35 people will eat 70 burgers in 20 minutes.

4.10    There is no vanishing money here.

The friends have paid 120/-, of which 100/- has been retained by the restaurant and 20/- has gone into the pocket of the dishonest steward.

Since the friends paid 120/- and they got back 30/- from the original 150/-, no money is missing.

The way the puzzle is worded misleads you into equating the amount got back by the three friends with the amount retained by the Steward.

4.11    This is a straightforward one. The numbers on the outer blocks increase in a clockwise direction by the amount of the number in the centre. Thus, the central number in the third figure will be 7

4.12    The two possible answers are 104 and 399.

In the first case, you will see from the first and the third figures that the central number is a simple sum of the three corner numbers. Thus, in the middle figure we add up the corner numbers:  56+36+12 = 104

In the second case, you will find that the numbers of the middle figure are the result of the multiplication of the corresponding figures in the first and the third figure. For example, 4*14=56. Thus, here we multiply the central numbers of the first and the third figures: 19*21 = 399

4.13    When you look closely, you will find that there are two alternating series here: +5 and -3. Hence the next two numbers in the series will be 33 and 17.

4.14    The solution to this is 32 =4+5

4.15    24 days. On the last day the monkey will climb 7 feet and climb out, so it will not slip back. Hence before this the monkey has to climb 23 feet which will take 23 days since its net climb rate is 1 foot per day.

4.16    The two possible solutions are given below.:

a) When we add the RHS of the previous line to the LHS of the next line, the equations balance. Hence in this case the answer will be 40.

1 + 4               = 5

2 + 5+5          = 12

3 + 6+12        = 21

8 + 11+21      = 40

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b) The equations balance when we express the LHS as:

 First Number + (First Number * Second Number).

Thus, in this case the equations will become:

1 + (1 x 4)      = 5

2 + (2 x 5)      = 12

3 + (3 x 6)      = 21

8 + (8 x 11)    = 96

Hence, the answer in this case will be 96