## Bank cashier mistakenly reversed cheque rupees with paise

Bank cashier mistake puzzle: Double the cheque amount less 20 paise received as Cashier interchanged paise with rupees. What were the cheque figures?

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

**Recommended time:** 20 minutes.

### Can you solve the bank cashier mistake puzzle mathematically?

Solve the puzzle mathematically with no trial and error.

No time limit for this part.

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, the legendary puzzlist.

### The Story of the Reverse Cheque Puzzle or Bank Cashier Mistake Puzzle

**Meaning of terms:** **Khokon** - pet call name for a young boy; **Dadu** - Grandfather;

This is an incident that had 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 the bank, and confidently encashed 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.”

As he was keen on mathematical puzzles, he understood the mystery.

And completed his instructions, “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 the 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 the cashier mistook the 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 in two variables. There would be an 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 the bank must be less than 100. If any paise figure equals or exceeds 100, 1 is added to the 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,

$x \leq 99$, and, $y \leq 99$.

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 the 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 solved first time by the natural method using math reasoning and a little trial and error. This is what I know as the Natural method of problem solving.

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

To keep 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. At the end, we would also show how the puzzle is solved naturally with reasoning and just a bit of trial.

### Bank Cashier Mistake Puzzle: 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 up 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, $y \gt 2x$ as $199 > 2\times{98}$..

And, $y-2x > 0$.

Simply speaking, the 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 Rupees by the cashier*is a little more than double the original cheque Rupee figure $x$. - This
**extra number of rupees**turned into paise (by multiplying with 100) and after deducting 20 and adding the paise received from the 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.

Understanding this mechanism of the mistake of the cashier and doubling the cheque amount is crucial for solving this puzzle quickly.

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

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

In the next step, 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, $y-x < 68$, both sides divided by 3 with RHS value taken as the larger 204.

**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 the actual value of $y$ were less than 99, the LHS will be further lower, so the inequality holds),

$2y-x < 167$.

Add 20 to both sides of the inequality,

$(2y-x)+20 < 187 < 200$, .................(4)

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

$(y-2x)<2$,

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

### Faster natural solution to Reverse Cheque Puzzle or Bank Cashier Mistake Puzzle

The linear equation is,

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

Also, $y -2x > 0$ from our previous initial analysis.

Comparing the increase of the LHS and RHS with the increase in this difference, you find that with every unit increase in $(y-2x)$, the value of LHS increases 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 the 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 the 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, I felt it to be excessively and unnecessarily complicated. So in this new version, I 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 learn all the time.

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