## Conditions to be evaluated not sequentially but strategically—Strategic logic condition analysis

In this fourth session on reasoning puzzles for Bank PO exams, the puzzle chosen is a floor stay puzzle that has fairly large number of conditions as well as is a three variable or three dimensional reasoning puzzle that is not easy. Finally though it has been a pleasure solving this apparently difficult puzzle. Majority of the basic and advanced patterns and methods in this topic area needed to be used in line with proven analytical strategy.

Before going ahead further, *you may refer to our earlier tutorial sessions and the solved reasoning puzzles with links given at the end of the solution*.

### Tricky Bank PO level floor stay Reasoning Puzzle

#### Problem description

In an eight-storied building, having floors numbered 1 to 8 from bottom to top, eight children A, B, C, D, E, F, G and H each live on a different floor not necessarily in the order of floor numbers. The children have different number of comic books with them 11, 16, 19, 25, 34, 41, 46 and 50, also not necessarily in the same order as their names or floors.

#### Conditional statements

- The child who lives on 6th floor has 25 comic books.
- One child lives between F and the one having 25 comic books.
- G lives below F on an even numbered floor.
- G does not have 25 comic books.
- Child having 46 comic books lives just above G.
- Two children Live between F and H.
- H lives below F.
- Total of number of comic books with D and H is a multiple of 4.
- Two children live between A and the child having 41 comic books.
- A lives above G.
- The one having 41 comic books lives above A.
- B has 34 comic books.
- The one having 11 comic books lives just above the one having 16 comic books.
- 1 child lives between C and E.
- The difference between the number of comic books with E and G is 6.

#### Questions

**Question 1.** Which of the following is correct?

- Two children live between D and the child having 11 comic books.
- G lives on the floor just below F.
- A has minimum number of comic books.
- E has 19 comic books.
- None of these

**Question 2.** On which of the following floors does D live?

- Floor 1
- Floor 3
- Floor 5
- Floor 4
- None of these

**Question 3.** Child having 50 comic books lives on,

- floor 7
- floor 4
- floor 2
- floor 1
- None of these

**Question 4.** Who lives just above A and with how many comic books?

- E with 25 comic books.
- C with 46 comic books.
- D with 19 comic books.
- B with 34 comic books.
- None of these

**Question 5.** Who lives two floors above B?

- F
- E
- C
- D
- None of these

### Solution to the Bank PO level tricky floor stay Reasoning Puzzle

#### Logic table representation

Following is the logic table that we will use for solving the problem.

First column of the three column logic table represents floor numbers and second column—the child living on a particular floor. The number of comic books with the child living on the particular floor is recorded in column 3. As usual, the floors are numbered from 1 to 8, and from bottom to top. 1 is the bottom-most floor and 8 is the top floor.

There are eight floors, eight children and eight numbers of comic books. It is a three variable or three dimensional assignment problem where our job is to place a unique combination of child name and number of comic books against each of the eight floors satisfying all the given conditions. This type of problem is also known as **assignment logic analysis problem,** where the three variables, Floor number, Child and Number of comic books with the child all are in **one to one relationship.**

When we process the conditional statements one by one, **not necessarily from beginning to end**, the conditions determine which child owning what number of comic books can occupy which floor. At the end of processing, no floor will have any uncertainty with respect to its occupancy. Each child will be placed on a distinct floor along with the right number of comic books.

As no cell will remain empty after we complete analyzing all the conditional statements, we call this form of logic table as Compact logic state representation or simply **Compact logic analysis table.**

The overall method of solution initially we named as collapsed column logic analysis technique; but you don't need to remember the names—just try to understand how and why the actions are taken and that will improve your ability to solve such problems. If you follow up by actually solving such problems a number of times with this grounding, may be in your own way, then only you will be able to solve practically any reasoning puzzle of this type quickly and confidently.

You may skip the following essential concepts and methods on reasoning puzzle problem solving if you are already aware of the same.

#### Essential concepts and methods on reasoning puzzle problem solving

* It is not necessary that the conditional statements are to be processed from beginning to end*. Usually that never happens. First step of any reasoning puzzle problem solving is to go through all the conditions and the description of the problem once. This is

**Initial problem analysis.**For large and complex reasoning puzzles, *this step is essential to understand the complexity of the puzzle, identify the promising statements, identify the primary barrier, if any and so on.*

Whatever be the problem though, at the start, **those statements are taken up for processing first** *that fill up up maximum number of cells with certainty*. This we call, * direct assignment first strategy*.

After a few cells get their values, the next strategy applied is * link search*, by which those statements are identified for processing that

*refer to any of the values of the already filled up cells.*

In case link search fails, the statements that refer to *multiple variable values of same type or different types with fixed positionally separated relation* are taken up to create what we call * temporary bonded member structures*.

If there are * two*,

*involving two (or more) values of same variable (or different variable), each of which we can place in specified positions without knowing which possible combination is the valid one, we create the two combinations as*

**but usually not more than two such possible combinations***. These are also temporary bonded structures but position of the values in each possible configuration formed is known without knowing which configuration is the correct one.*

**possible configurations by combining these temporary combinations with existing certain combination**When positions of temporary bonded structure members are not known, or possible positions are too many, we keep the temporary bonded structure separate from the main logic table. These are called * floating bonded member structures*.

*Temporary bonded member structures are important because further conditions refer to variable values in the structures and make them more certain.*

Let us now get down to the main task of processing the conditional statements with the **sole aim of filling up the empty cells as quickly as possible without any confusion and with complete certainty.**

#### Solution Stage 1: Strategy 1: Direct and certain assignment first

In any assignment puzzle, **Direct certain assignments in the beginning is a "highly preferred" priority**. Without such a direct certain assignment of a variable value to a position, we won't be able to fill up any cell of the logic table in the beginning, and further steps will be difficult to carry out.

We qualify this strategy with the additional property of a statement first to be executed as one with **direct and certain assignments of maximum amount or maximum number of cells filled up with certainty.**

We select such a statement and process it first to fill up maximum number of empty cells at one go. The advantage is, **the more we fill up the empty cells easier does it become to fill up the rest, as uncertainty reduces.**

Following this strategy, we select first **Statement 1.** "The child who lives on 6th floor has 25 comic books." By this statement, 25 is placed against floor 6 under comic books column with **certainty**. Placing against a floor the child on the floor or the number of books with the child living on the floor is equally important and has no particular priority in general.

**Finding no more direct assignment** we search for **a statement that refers to comic book number "25" at floor 6, **the only value placed with certainty till now. This is use of link search technique and we locate **Statement 4.**"G does not have 25 comic books." We record this negative but certain information as "not G" under column of Child against floor 6.

The state of the logic table now is shown below. Out of eight numbers, 25 is crossed out to keep track of what values are left to be assigned.

#### Solution Stage 2: Searching for temporary bonded structures and forming possible configurations

As there are no more statement to directly assign any value, we have no other option now than to look for the *next most valuable statement* or group of statements that create *temporary bonded structures separated by fixed number of floors* or *even create possible configurations*.

Accordingly we select three statements together,

**Statement 2. **"One child lives between F and the one having 25 comic books",

**Statement 3.** "G lives below F on an even numbered floor", and

**Statement 5.**"Child having 46 comic books lives just above G."

We first identify Statement 2 by its **link reference to already placed value 25.** But additionally, the Statement 2 creates the **three-floor spanning bonded structure** of F and comic book number 25 with one floor in between. As this relationship is a *positionally fixed one, spanning three floors*, while trying to place this on the logic table at any time, we would have less number of possible ways to place. Larger the size of the bonded structure lesser will be the possibilities to place it. In other words, * uncertainty of placement will decrease, and placement will be more certain* as the size of such a temporary bonded structure increases.

In Statement 3, G is linked to F who appears in Statement 2. Also in Statement 5, a second short length bonded structure is created which also has the additional property of link reference to G appearing in Statement 3. This is **chained link search in action.**

So we analyze these three statements together, form the possible bonded configuration and try to place the possible bonded configuration against floors. The result is as below. First try to verify what we have done and understand why. We will explain shortly the reasons.

There are two possibilities in Statement 2, F can either be below 25 at floor 6 or above it. If F is below 25 at floor 6, by the next Statement 3, G can occupy only floor 2 and so F goes to only available floor 4 with 46 against floor 3 by Statement 5.

But if F is above 25 at floor 6, though it can occupy only floor 8, G can be placed either in floor 2 or floor 4. *We don't like the situation of creating three possible configurations,* but in this case accept it because the configurations are simple two column uncertainties with fairly occupied cells. Hopefully next few statements will be able to eliminate some of the possible configurations by conflict.

**Important to note**

At this stage any one of three three pairs of column values of "Child-Comic Book" may be the correct one, but we don't know which will finally turn out to hold the solution. Mark that the main table is also a possibility at this stage, not a certainty. We can name it, if you want, as Possibility 0. Also note that the existing certain values in the main table have been repeated in the other two possibilities while forming them. For correctness of analysis in subsequent stages, this action is important.

**Solution Stage 3: Achieving certain placements in the three possibility configurations**

At this stage we choose those statements that either help to cancel a possibility configuration by proving it to be invalid, or creates certain placements in each of the three configurations. In other words, at this stage we won't allow any more increase in number of possible configurations and only move towards more certainty.

We select **Statement 6.** "Two children Live between F and H", and **Statement 7.** "H lives below F", because both together achieves certain placements in each of the three possibilities. **Link reference** to existing occupied value of F helped.

Interestingly, earlier, without the three possibilities with more number of occupied cells, these statements won't have been able to achieve these certain placements. *Think over for a moment.*

The logic table at this stage is as below.

#### Solution Stage 4: Cancellation of a possibility configuration by conflict

The situation is favorable now for elimination of one or both of the possible configurations by positional conflict because of fairly high occupancy of the cells in each configuration.

What type of statement should we choose now?

*A statement that forms the largest temporary bonded structure* would have the highest potential to achieve this goal. In this attempt we might choose more than one statement that are linked together by reference of existing values.

As thought out, we select **Statement 9**. "Two children live between A and the child having 41 comic books", as it spans four floors—A and 41 are separated by two floors. Automatically two more statements, **Statement 10.** "A lives above G", and **Statement 11**. "Child having 41 comic books lives above A", are grouped together in a more populated bonded structure because of link references. This temporary bonded structure with 41 and A separated by two floors and A below 41 is shown labelled as **TBS** in the figure below.

We find this combination with A above G and three floors below 41 to **violate** **Possible configuration 1**—no way can we place A in this configuration and so this **Possibility 1 is invalid**. We cross out this combination and go on to place the values in both remaining configurations.

After this encouraging result, we identify **Statement 12.** "B has 34 comic books" that also achieves certain placements in both the configurations.

The logic table is shown below.

#### Solution Stage 5: Cancellation of another possibility by conflict and final solution

Again we search out **Statement 14.** "1 child lives between C and E", because of its three-floor long temporary bond. We can't place C or E by this condition in possible configuration 2, making it invalid. We cross out this combination and go on to record "C/E" against floor 6 and "E/C" against floor 8 in the only remaining original configuration, indicating that either of C or E will occupy these two floors but we don't know yet for certain who belongs to exactly which floor.

This recording of C and E against two floors is important **as it blocks two positions with certainty**.

A **bit of theory ahead** that you may skip.

We call this structure a

Cycle, and the process, cycle formation. Cycle is a powerful structure to have, though we don't use its full power here. With two members in a cycle we have two degree uncertainty—if just one of the possibility becomes certain, automatically its counterpart also becomes certain by possibility cancellation.

Let us take up our analysis again.

*Because of the blocking of the two out of three remaining cells we gain immediately a certain placement of D against the only remaining floor 3*. Note that we achieve this certain placement without the help of any additional condition. This type of certain placement we call "**Placement by exclusion**" by principle of exclusion.

Interestingly, as we place D on floor 3, 46 gets automatically assigned to D, and we get the opportunity to process **Statement 8.** "Total number of comic books with D and H is a multiple of 4". Among the remaining numbers 11, 16, 19 and 50, only 50 satisfies this condition (50+46=96, you need to know a bit of maths also), and so H gets 50.

Only 11, 16 and 19 remaining, the next **Statement 15.** "The difference between the number of comic books with E and G is 6" gives us not only 19 for G but also E against floor 6 with 25 and by **possibility cancellation** C against floor 8.

Last **Statement 13.** "Child having 11 comic books lives just above the one having 16 comic books" places 11 against floor 5 and 16 against floor 4 with certainty.

No more cell is empty—all are filled up. Assignment is complete.

The following is the solved logic table. Statement 14 is highlighted as it played crucial role in eliminating the **Possibility 2**.

Now we are ready to answer the questions.

#### Answers to the questions

**Question 1.** Which of the following is correct?

Answer 1. Option 3: A has minimum number of comic books.

**Question 2.** On which of the following floors does D live?

Answer 2. Option 2: Floor 3.

**Question 3.** Child having 50 comic books lives on,

Answer 3. Option 4: floor 1.

**Question 4.** Who lives just above A and with how many comic books?

Answer 4. Option 1: E with 25 comic books.

**Question 5.** Who lives two floors above B?

Answer 5. Option 5: None of these.

### Comment

Though this is a tricky floor stay reasoning puzzle with a fairly large number of logic conditions, condition analysis and strategic use of proven patterns and methods such as direct assignment first, link search by reference, temporary bonded structure formation and possible configuration formation helped us to reach the solution quickly in a few steps. Elementary logic analysis formed the basis of identification and use of every pattern and its method.

The **problem is well-formed**, as even with pattern based quick solution, all conditional statements needed to be used. *None of the statements turned out to be superfluous* which is not usually the case with this type of puzzles with large number of conditions.

### End note

Solving reasoning puzzles does not need knowledge on any subject—it is just identifying useful patterns by analysis of the problem and using appropriate methods. It improves problem solving skill, because patterns and methods lie at the heart of any problem solving.

### Other resources for learning how to discover useful patterns and solve logic analysis problems

#### Einstein's puzzle or Einstein's riddle

The puzzle popularly known as Einstein's puzzle or Einstein's riddle is a six object set assignment logic analysis problem. Going through the problem and its efficient solution using collapsed column logic analysis technique in the session * Method based solution of Einstein's logic analysis puzzle whose fish* should be a good learning experience.

#### Playing Sudoku

As a powerful method of * enhancing useful pattern identification and logic analysis skill*, play

**Sudoku**in a controlled manner. But beware, this great learning game, popularly called Rubik's Cube of 21st Century, is addictive.

To learn how to play Sudoku, you may refer to our **Sudoku pages***starting from the very beginning and proceeding to hard level games.*

### Reading list on SBI PO and Other Bank PO level Reasoning puzzles

#### Tutorials

**How to solve SBI PO level logic puzzles in a few simple steps 1**

** How to solve SBI PO level logic puzzles in a few simple steps 2 **

**How to solve SBI PO level family relation problems in a few simple steps 3**

**How to solve SBI PO level floor stay Reasoning Puzzle in a few confident steps 4**

**How to solve high level circular seating reasoning puzzles for SBI PO in confident steps 5**

**How to solve high level hard two row seating reasoning puzzles for SBI PO in confident steps 6**

**How to solve high level circular seating arrangement reasoning puzzles for SBI PO quickly 7**

**How to solve high level nine position circular seating reasoning puzzles for SBI PO quickly 8**

**How to solve high level box positioning reasoning puzzle for SBI PO quickly 9**

#### Solved reasoning puzzles SBI PO type

**SBI PO type high level floor stay reasoning puzzle solved in a few confident steps 1**

**SBI PO type high level reasoning puzzle solved in a few confident steps 2**

**SBI PO type high level reasoning puzzle solved in a few confident steps 3**

**SBI PO type high level circular seating reasoning puzzle solved in confident steps 4**

**SBI PO type high level hard reasoning puzzle solved in confident steps 5**

**SBI PO type high level one to many valued group based reasoning puzzle solved in confident steps 6**

**SBI PO type high level hard two in one circular seating reasoning puzzle solved in confident steps 7**

**SBI PO type hard facing away circular seating reasoning puzzle solved in confident steps 8**

**SBI PO type high level four dimensional reasoning puzzle solved in confident steps 9**

**SBI PO type hard two row seating reasoning puzzle solved in confident steps 10**

**SBI PO type high level floor stay reasoning puzzle solved in confident steps 11**

#### Solved reasoning puzzles Bank PO type

**Bank PO type two row hybrid reasoning puzzle solved in confident steps 1**

**Bank PO type four variable basic assignment reasoning puzzle solved in a few steps 2**

**Bank PO type basic floor based reasoning puzzle solved in a few steps 3**

**Bank PO type high level floor stay reasoning puzzle solved in quick steps 4**