Thinking with Numbers - Part 2

Lesson Objective

We already covered 3 different types of puzzles in the previous lesson of thinking with numbers. This lesson will be the continuation of the previous lecture and we will discuss some more puzzles in this lecture. This is a self-paced class. So, this will also be a fun lesson like the previous one.

Numbers are crazy and did you hear them talk…? 😉

Takeaways from the previous lesson

The most important takeaways from the previous lesson were:

There is no single predefined approach to solve all problems, whether they might be fun puzzles or programming problems you will be solving in the future
You should be able to think mathematically in terms of factors, multiples, LCM, HCF, etc. when dealing with numerical problems
The process of elimination of possibilities that will not be correct or feasible for our solution is very important.

Introduction to the Lesson

Today, we will be solving some number puzzles which will mainly focus on your ability to guess the possible permutations and combinations of digits that can form the solutions to these problems.

This will involve first of all your ability to observe and gather as much information from the problem as you can and accumulate all this information in your head to use it for solving the actual problem.

The specific puzzles we will discuss today are these three:

Digit Challenge
Crack the Code
Kakuro Puzzle

1. Digit Challenge

We will be looking at some fill in the blanks type puzzles here. The puzzle will consist of a simple mathematical equation and you will be given some digits and you have to use only these digits to fill up the equation such that the equation is correct. These puzzles are designed to test your numerical comprehension and whether you are able to think with numbers at a basic level.

Problem Statement: Correct the equation given only using the digits given

The first and important step to approach any problem is to understand the problem and understand the constraints that are specified by the problem/puzzle.

For these problems, you have to keep in mind that

Each digit can be used only once
You can only use the digits provided
There may or may not be more than one solution to the problem

Approach to Digit Challenges

Some important factors that can be applied in these problems (not specific to these problems only) and you might have discovered some of them already are:

Multiplicative Identity(i.e. 1) and Additive Identity(i.e. 0)
Precedence of Operations: In the example, only one operation is being performed but there can be multiple operations as well.
If a predefined result is ending with 0 or 5, we can conclude that it is a product of number 5
Try to consider divisibility rules and various factors and multiples of these digits.

Let’s discuss the sample problem we have taken, we can see that we have to use all the six digits provided to solve the problem. Then, we can try to brainstorm some combinations, since the RHS will of course be the largest term of the three. I put a 9 there just to observe other possibilities. Then, we can try using the other digits as the second digit with 9 in RHS. First, we try to put 0 with 9 to get 90. Now, check if we can form 90 with 2 numbers made from the digits 1, 2, 7, 8.

Step 1: We can see that we have to use all the six digits provided to solve the problem. Then, we can try to brainstorm some combinations.

Step 2: Since the RHS will of course be the largest term of the three, I put a 9 there just to observe other possibilities. Then, we can try using the other digits as the second digit with 9 in RHS.

Step 3: First, we try to put 0 with 9 to get 90. Now, check if we can form 90 with 2 numbers made from the digits 1, 2, 7, 8.

If you observe closely, you can see a possibility….

Yes, you are correct we can use 78 and 12, and 78 + 12 = 90

Thus, our problem is solved.

Note: This problem has one other solution as well, try finding it.

Check the Hint!

This time start with 8 as tens digit in RHS

Check the solution!

The solution is 7 9 + 0 2 = 8 1

Follow Up

You can try to solve some other problems involving other arithmetic operations as well. Remember the tips but do not constrict your thinking with them.

Correct the equation given only using the digits given:

Check the solution!

( 4 / 2 ) + 6 + 7 = 1 5

Correct the equation given only using the digits given:

Check the solution!

( 3 / 4 ) x 2 x 6 = 9

2. Crack the Code

We will be looking at some fill in the blanks type puzzles here as well. This will be a fun puzzle and you will find that as you identify the hints in problems and connect the dots you move closer to the solution to problems. You will be given a 3-digit puzzle in the samples we will discuss, you can find 4-digit puzzles online if you find these puzzles interesting.

Problem Statement: You are given that a 3-digit code can unlock the lock along with some hints to guess the 3 digit. Your task is to identify these 3 digits in the code and you will be given a few wrong combinations with the details of why they are incorrect or correct.

We will take the following sample problem for better understanding:

Approach to Crack the Code

So, we have been given three incorrect combinations for the lock. You will use these combinations to infer the correct code for the lock.

Using the 1st combination

In the first code, we can see that it has been specified that two numbers are correct and they are in their correct positions as well. So, one of the pairs out of 1,6; 1,5; 6,5 are correct and the digits in that pair are in their correct spot as well.

Using the 3rd combination

At the same time, we can also check out the last combination to see what’s not possible and then use that information to cancel out the digit that is wrong. We observe that the numbers 2,6 and 4 cannot be in the code. So, we can further eliminate two of the pairs which we considered earlier because they contain 6. So, the number 1 & 5 in the 1st combination are correctly placed in their spots. Now, we only need to find the middle digit.

Using the 2nd combination

In the 2nd combination, we observe that two numbers are correct but they are at incorrect positions. We surely know that one of those numbers is 5. So, the middle digit can be 3 or 8. If you inspect a bit more, you will notice that 3 cannot be that number because it has been stated that the correct digit is at an incorrect position but looking at 3 it is in the middle (which is the correct position according to our observations). Hence, the middle digit is 8 and congratulations you’ve cracked the code.

The correct code is: 1 8 5

All in all, you have to keep in mind these points to solve these problems:

Try to identify the digits that are in the code by using the information to formulate a set of possible combinations.
Try to eliminate options from these combinations by identifying the information about wrong digits.

Follow Up

You can take insights from the process we discussed above for solving the sample puzzle and solve some more of these puzzles given below.

If you feel 3-digit codewords are easy for you, try giving the 4-digit one a try as well.

Find the 3-digit code using these hints provided

Check the solution!

Solution: 1 6 4

Find the 4-digit code using the hints provided

Check the solution!

Solution: 3 8 4 1

3. Kakuro Puzzle

Kakuro is a logic puzzle that is often referred to as a mathematical version of the crossword puzzle you must’ve seen in newspapers. The puzzle consists of a grid with cells and some of these cells are empty and some are filled with 1 or 2 numbers. These cells with the 1 or 2 numbers tell us how to fill the empty cells and a standard Kakuro puzzle has a unique solution. And because the solution is unique, the puzzle will only involve investigation of combinations and not various permutations like Sudoku. Let’s understand the puzzle in detail.

Problem Statement: Fill in the empty cells in the given grid using the instructions below:

We have already discussed the importance of understanding the problem/puzzle and the constraints that are specified before rushing in to approach the problem. So, let’s understand what exactly has to be filled in the empty cells. As you might have observed, there are three types of cells in the puzzle and those are:

Barred cells: These are the cells not involved in the puzzle.
Filled cells(consisting of one or two numbers)
Empty Cells: These are the cells that have to be filled to solve the puzzle.

The filled cells contain a diagonal slash which divides them into two halves, and a number is there in one or both halves(known as a “clue”). You can see that the empty cells form up a system of rows and columns. Your task in these puzzles is to fill these rows and columns with digits from 1 to 9 such that none of them is repeated in that row and column.

For a row, the sum of the numbers in that row must be equal to the number specified to the immediate left of the start of that row of cells
For a column, the sum of the numbers in that column must be equal to the number immediately above the start of that column of cells

Approach to solve Kakuro

Let us now try to solve the sample puzzle we have taken. Here, first of all you can observe that there are a total of 4 rows and 4 columns. Now, to start filling the cells, you should consider the rows that have only one possible combination to fill them. For example, the first row requires a sum of 3 from two cells which is only possible with the numbers 1 and 2.

Now, we have to decide the order in which these numbers will be inserted into the cells. To decide that, consider other trivial columns like the right-most column which requires 2 cells to fill up to 4 and the only possible pair of digits that can be used are 1 and 3. Because the digits cannot repeat in their row and column. You must’ve guessed the correct order for topmost row and rightmost column i.e. 2, 1 and 1, 3 respectively.

Now, let's consider the second row(with clue 6), we see that the two empty cells can only be filled with1 and 2. To decide the order, we look at the second-last column (with clue 6), it already has a 2, so the cells will be filled as shown: