The Art of Confusing

The Art of Confusing.

Riddles, Puzzles, and Magic thrive on the confusion that it causes in the mind of the reader, the onlooker. The surprise and entertainment it provides someone comes from the revelation of a fact that is hidden from the first glance or expectation of the person experiencing it. Here are some mathematical cases that may give you one impression at first glance and another one on a closer look.

BODMAS

1.Question:

Is it right or wrong? 230-220*.5 = 5!

If you do note notice the exclamation mark at the end you are likely to treat it wrong or think that it is due to giving a higher precedence to subtraction than to multiplication, like (230-220) *.5 = 5

If you are careful to note the exclamation (factorial symbol) it depicts correct evaluation like 230 – (220*0.5) = 120 = 5!

A subtle difference to confuse you.

This I received first time in my whatsapp from our magician friend Pradeep.

At the first glance it is likely to be confused as (230-220) *.5 = 5 (taking ‘!’ as just an exclamation) and judged as right.

But for those who consider arithmetic operations and their precedence, it is actually to be evaluated as 230 - (220 * .5) = 230 – 110 = 120 = 5! and deemed correct. (taking ‘!’ as symbol for factorial)

It is here that we need to take note of the exclamation mark and that the RHS is factorial of 5 which is 120.

A good example of a confused evaluation.

But what caught my interest is possibility of this doubly interpretable expression with other numbers. Is there any other like this?

First, by trials I found that 40-32*.5=4! Is similar.

In general, for any n (better be > 3 to avoid trivial cases) (2 * n! - 2n) – (2*n! - 4n ) * 0.5 = 2*n! – 2n – n! + 2n = n!

((2 * n! - 2n) – (2*n! - 4n ) ) * 0.5 = n! – n - n! +2n = n

So the answer to our question is in the affirmative. There are as many cases as there numbers.

What happens if we take n as 6?

1428 – 1416 * 0.5 = 1428 – 708 = 720 = 6!

(1428 – 1416) * 0.5 = 12 * 0.5 = 6

The cases above are those with 0.5 as multiplication factor. There are others with different multiplication factors too.

12 – 8 * 0.75 = 3!

(12-8) * 0.75 = 3

* * * * * * * * *

2. Right way of distribution.

This one I collected from a book on the mathematical Olympiad.

A, B and C are friends living together sharing the household chores and expenses equally among them.

One day, they are setting out for collecting firewood for their use. C, who is unwell opts for taking rest and offers to compensate the other two for the firewood they bring.

In the evening, A brings 5 bunches of firewood and B, 3 bunches.

C gives 8 rupees for A and B to share among themselves. A takes 5 rupees and B takes 3 rupees. 

Question: Is the distribution proper?

Answer: No.

Explanation: The firewood is not for the exclusive use of A and B but all three. We can estimate 8/3 bunch will be used by each. C need not compensate for what others use. His compensation is for only what he uses, viz. 8/3 bunch. So his compensation of 8 for 8/3 bunches works out equivalent to 24 rupees for 8 bunches, ie. 3 rupees for each bunch. So what A has brought is worth 15 rupees and that of B is worth 9 rupees. If we subtract their usage of firewood worth 8 each, what is due to A comes to 15-8=7 rupees and that of B comes to 9-8=1 rupees.

A should get 7 rupees and B should get 1 rupee.