## Don't forget to use all the conditions in a problem situation

### The puzzle brief

Twice the amount written on cheque remained after spending 20 paise from amount received from bank. Cashier mistakenly interchanged paise with rupees. What were the cheque figures?

**Information:** 100 paise made 1 rupee.

**Recommended time:** 20 minutes.

#### Part 2 of the puzzle

Solve the puzzle mathematically with no trial and error.

No time limit for this part.

*But do continue even if you exceed the limit, the puzzle is not that difficult and it is highly enjoyable.*

As you start working on this classic puzzle, let us go ahead with the story of the puzzle and then the solutions. Puzzles are made interesting by the story and this story is described exactly as my friend recounted to me.

Much later, we came to know that this puzzle was created by none other than Henry Dudeney, to many as the greatest puzzler of all time.

### The Story of the Reverse Cheque Puzzle

This is an incident that happened many years ago.

One day Bhabababu, a fairly wealthy man, decided to test the world-worthiness of his young grandson and asked him, “Khokon, can you get this cheque cashed in my bank? You know where it is. Last month I took you with me.”

Khokon was a smart young lad full of enthusiasm. He said proudly, “Yes Dadu, of course.”

So he took the cheque from his Dadu, went to bank, and confidently cashed the cheque.

At home when he handed over the money to his Dadu, Bhabababu counted the money twice. He couldn’t believe it. With deep surprise mixed with a tinge of anxiety he asked, “Khokon, what did you do with the money?”

Khokon replied apprehensively, “Why Dadu, I have spent only 20 paise from the money I got from the Bank as tram fare. Did I do anything wrong?” His Dadu replied, “No Khokon, you didn’t. The cashier did.”

Being keen on mathematical puzzles, he understood the mystery.

Continuing, he instructed Khokon, “The money that I got is exactly double of what I wrote in the cheque. By mistake the cashier had given you the rupees written as paise and paise written as rupees in my cheque. Here, take this extra money. I have added 40 paise for your two-way tram fare. Go quick Khokon. The cashier might be in trouble!”

**Two questions:**

- What was the cheque amount, and how did you get the answer?
- Can you deduce it mathematically, meaning, can you produce a deduction that is full of equations and mathematical procedures?

**Information:** 100 paise made 1 rupee.

### Solution to Reverse cheque puzzle or the puzzle also known as Bank cashier mistake puzzle

#### Problem definition

Let $x$ be the cheque rupee figure, and $y$ be the cheque paise figure.

As finally received amount is double the cheque amount, it is,

$2(100x+y)$.

This equals the amount given by the cashier minus 20 paise. As cashier mistook rupee figure for paise and paise figure for rupee, $x$ and $y$ were interchanged.

Arriving at the finally received amount from the amount given by the cashier, the finally received amount is also then,

$(100y+x)-20=2(100x+y)$.

This is a single equation on two variables. There would be infinite number of pairs of values for $x$ and $y$ satisfying the equation unless there are any other restrictions on the values. This is a simple mathematical truth.

In simple words, this equation is unsolvable by itself.

**Domain condition:** In real world transactions, the paise written in cheque or received from bank must be less than 100. If any paise figure equals or exceeds 100, 1 is added to Rupee figure and the rest paise forms the Paise figure.

Mathematically then, especially because $x$ and $y$ both served the role of paise and rupee,

$x \leq 99$,

$y \leq 99$.

These two inequalities along with the single linear equation in $x$ and $y$ should give us the unique solution.

Mathematics needed to solve the puzzle would largely be** inequality analysis and inequality algebra** that we are not used to normally. That's why mathematical solution of this puzzle seems to be difficult.

#### Comment

You can solve this puzzle by a little bit of reasoning and trial and error. It shouldn't take long. In fact i and my friend both solved first time by that natural method. Maths is used in natural method of problem solving, but more of mathematical reasoning and trial and error it contains.

*It is fun solving any problem naturally, in a way that is easy to understand by most people. A little trial and error is part of the game of natural method of problem solving.*

To retain the flavor of the natural way of solving this classic little puzzle, we would describe the meaning of mathematical operations or expressions in plain English time to time as well as take the first opportunity to show how it is solved naturally near the end with just a bit of trial.

Upshot is, though we would proceed to solve mathematically with occasional explanation in plain language but would also show the natural solution when the time comes.

### Problem solving: Strategic approach to narrow down feasible values by inequality analysis and inequality algebra

Three relations on cheque Rupee $x$ and Paise $y$ are,

$(100y+x)-20=2(100x+y)$,

$x \leq 99$, and,

$y \leq 99$.

Let's take on the first equation as it contains maximum potential,

$(100y+x)-20=2(100x+y)$,

Or, $98y=199x+20$,

Or, $y=\displaystyle\frac{199}{98}x+\displaystyle\frac{20}{98}$.

So, $y \gt 2x$.

And, $y-2x > 0$.

Simply speaking, value of cheque Paise is more than double the value of cheque Rupee figure.

This you would also realize with not much of a difficulty. The expenditure of 20 paise would be possible only when,

The

cheque Paise $y$ after being mistaken as Rupee by the cashieris a little more than double the original cheque Rupee figure $x$.This

extra number of rupeeturned into paise (by multiplying with 100) and after deducting 20 from and adding the paise received from bank becomes equal to double the cheque Rupee figure $x$ that was taken by the cashier as paise.

How the mistake had happened and the final received amount became double the cheque amount is easy to visualize from the following figure.

Perhaps with this understanding you are already well on the way to find the solution. In fact, understanding this mechanism of the mistake of the cashier and doubling the cheque amount is crucial for solving this puzzle quickly.

**If you like to see the natural quick natural solution** jump to it directly by clicking **here.**

Coming back to our mathematical solution, we realize that the value of this extra, $(y-2x)$ is the key to solving the puzzle.

By trial you can easily get its value. But our job is to solve the puzzle without trial, purely by math.

With the realization, the equation is normalized with respect to the coefficients of $x$ and $y$ and expressed in terms of $(y-2x)$,

$(100y+x)-20=2(100x+y)$,

Or, $100(y-2x)=(2y-x)+20$.............(1)

As $x$ and $y$ are both less than 100,

$(2y-x) < 200$....................(2)

From the equation 1, as $\text{RHS} < 300$, in LHS,

$y-2x < 3$...........................(3)

Add equation (2) and equation (3),

$3y-3x< 203$,

Or, $y-x < 68$.

**Note:** adding both LHSs smaller than the respective RHS will be less than sum of the two larger RHSs.

Add maximum possible value 99 of $y$ in RHS for a $y$ in LHS (if actual value of $y$ were less than 99, the LHS will be further lower, so the inequation holds),

$2y-x < 167$.

Add 20 to both sides of the inequation,

$(2y-x)+20 < 200$.................(4)

With this RHS of equation (1) less than 200, in LHS

$(y-2x)=1$.............................(5)

Substitute this value in (1),

$(2y-x)=80$...........................(6)

Multiply equation (5) by 2 and subtract the result from equation (6) to eliminate $y$,

$3x=78$,

Or $x=26$, and $y=53$.

**Answer:** Original cheque amount was Rupees 26 Paise 53.

### Quicker natural solution

The linear equation is,

$100(y-2x)=(2y-x)+20$.............(1)

Also, $y -2x > 0$.

Comparing the increase of the LHS and RHS with increase in this difference, you find that with every unit increase in $(y-2x)$, the value of LHS increase by another hundred.

As $x$ and $y$ are less than 100, the RHS won't be able to keep up with this galloping increase of the LHS, if the value of $(y-2x)$ is large.

Best is to start with the lowest figure of $(y-2x)=1$.

You get the solution immediately with this value.

### Actually what happened

Cheque amount being Rupees 26 Paise 53, cashier interchanged the two and with $y-2x=1$, the extra rupee equivalent to 100 paise acted as the overflow (just like in addition of two numbers). After deduction of 20 paise, the overflow became 80 paise. Add this with 26 paise given by the cashier. Result becomes 106 paise, just the double of 53.

### Remarks

When realization dawned that $(y-2x)$ can have values either 1 or 2, just test with 1 for success or 2 for invalidity, and the answer will be immediately known. But that involves trial.

Our take was to avoid trial altogether. That's why this boring long mathematical rigmarole.

**Recommendation:** Try to improve the solution.

### Changelog

**This major revision:** On 2nd of Jan, 2021.

After first posting the solutions three years back, when reviewed the solutions today, we felt it to be excessively and unnecessarily complicated. So in this new version, we decided to drop all the excesses and make the solutions as short and easy to understand as possible.

You might be able to improve it further. You know, we are not all knowing, rather we continue to know all the time.

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