Rules of the game: 1. No line may cross another line. 2. No line can cross the black bands (see description at the bottom of the puzzle). 3. Only ONE line can cross the band with stripes. 4. The wavy dotted band MUST be crossed by at least THREE different connecting pathway lines. 5. Respecting the previous points, every pathway line should be the SHORTEST possible.

The following formula - named Zeller's Rule - allows you to calculate a day of the week for any date:

F = k + [(13 x m-1)/5] + D + [D/4] + [C/4] - 2 x C

k is the day of the month. Let's use January 27, 2024 as an example. For this date, k = 27.
m is the month number. Months have to be counted specially: March is 1, April is 2, and so on to February, which is 12 (this makes the formula simpler, because on leap years February 29 is counted as the last day of the year). Because of this rule, January and February are always counted as the 11th and 12th months of the previous year. In our example, m = 11. 
D is the last two digits of the year. Because of the month numbering, D = 23 in our example, even though we are using a date from 2024.
C stands for century: it's the first two digits of the year. In our case, C = 20.

Now let's substitute our example numbers into the formula:
F = k + [(13 x m-1)/5] + D + [D/4] + [C/4] - 2 x C
= 27 + [(13 x 11-1)/5] + 23 + [23/4] + [20/4] - 2 x 20
= 27 + [28.4] + 23 + [5.75] + [5] - 40
[dropp every number after the decimal point]
= 27 + 28 + 23 + 5 + 5 - 40 = 48. 
Once we have found F, we divide it by 7 and take the remainder (if the remainder is negative, add 7). A remainder of 0 corresponds to Sunday, 1 means Monday, etc. For our example, 48 / 7 = 6, remainder 6, so January 27, 2024 will be a Saturday.

By means of the tables below, you can determine the day of the week for specific dates that might be of your interest. Just follow the easy instructions. First, locate the number at the intersection of the month column and day row in the table 1 (example: the column for June and the row for day 16th cross at the number 6). Then, locate the number at the intersection of the year column and century row in the table 2 (example: the column for year 77th and the row for century 18th cross at the number 3). Finally, locate in table 3 the letter at the intersection of both numbers you've found (here, numbers 6 and 3 cross at D). The letter D corresponds to Saturday (if the day of the week you're looking for is in January or February of a leap year, you have to bring forward of 1 day the result).

- G. Sarcone

• Nothing is as difficult as it looks.
• Nothing is as easy as it looks.
• Read the instructions carefully in order to avoid giving the right answer to the wrong puzzle.
• Human languages were not 'designed' for logical clarity. Remember that each topic within math (or within any field) has its own tricky phrases.
• Recognize clearly what you have to search for, what data is useful and what are the relationships between the searched result and the data. 
• Misdirection is very common in puzzles. Expert puzzle authors always try to lead people's minds along false trails.
• Elements that appear to have no relevance may be fundamental and vice versa.
• Omissions (what is not written or said) are sometimes as important as the instructions given.
• Beware of subconscious restrictions and mental blocks, try to depart from the norms and stereotypes and look beyond the boundaries of a problem.
• Break a complex puzzle into smaller, manageable parts.
• Ensure that the puzzle has been considered from all points of view.
• With logic, correlated elements or events are not necessarily related. Some could be just coincidence.
• Likewise, correlated events may have a common cause. In fact, some correlations may not be relations between cause and effect but represent two effects of some other cause.
• Either of two correlated events may cause the other.
• Sometimes, to understand a problem, you'll need to usecollective intelligence. Do not hesitate to ask someone else to help you.
• Use your intuition and deliberately try to find other ways to solve a puzzle.
• Sometimes, the solution is what causes the problem. Avoid finding solutions to nonexistent problems, and never fix what is not broken.
• Enjoy making mistakes. Regard them as important and fruitful stages on the road to success.
• Be persistent, the joy of achievement is greater when it is preceded by a tough creative effort!
• Puzzles always have one, several or no solutions.

GATE 2009 Question Paper - Part 1
1. Which one of the following is NOT necessarily a property of a group?
a. Commutativity
b. Associativity
c. Existence of inverse for every element
d. Existence of identity
2. What is the chromatic number of an n-vertex simple connected graph which does not contain any odd length cycle? Assume n>= 2
a. 2
b. 3
c. N-1
d. N
3. Which one of the following is true for any simple connected undirected graph with more than 2 vertices?
a. No two vertices have same degree
b. At least two vertices have the same degree
c. At least three vertices have the same degree
d. All vertices have same degree
4. Consider the binary relation R = {(x,y),(x,z),(z,x),(z,y)} on the set {x,y,z}. Which one of the following is true?
a. R is symmetric but not antisymmetric
b. R is not symmetric but antisymmetric
c. R is both symmetric and antisymmetric
d. R is neither symmetric nor antisymmetric
5. (1217)8 is equivalent to
a. (1217)16
b. (028F)16
c. (2297)10
d. (0B17)16
6. What is the minimum number of gates required to implement the boolean function (AB+C) if we have to use only 2-input NOR gates?
a. 2
b. 3
c. 4
d. 5
7. How many 32K X 1 RAM chips are needed to provide a memory capacity of 256K bytes?
a. 8
b. 32
c. 64
d. 128
8. A CPU generally handles an interrupt by executing an interrupt service routine
a. As soon as an interrupt is raised.
b. By checking the interrupt register at the end of the fetch cycle
c. By checking the interrupt register after finishing execution of the current execution
d. By checking interrupt register at fixed time intervals
9. In which one of the following page replacement policies, Belady’s anomaly may occur?
a. FIFO
b. Optimal
c. LRU
d. MRU
10. The essential content(s) in each entry of a page table is/are
a. Virtual page number
b. Page frame number
c. Both
d. Access right information
11. What is the number of swaps required to sort n elements using selection sort, in the worst case?
a. Ѳ(n)
b. Ѳ(n log n)
c. Ѳ(n2)
d. Ѳ(n2 log n)
12. S -> aSa bSb a b
The language generated by the above grammar over the alphabet {a,b} is the set of
a. All palindromes
b. All odd length palindromes
c. Strings that begin and end with the same symbol
d. All even length palindromes
13. Which of the following statement(s) is/are correct regarding Bellman-Ford shortest path algorithm?
P:always finds a negative weighted cycle, if one exists.
Q:finds whether any negative weighted cycle is reachable from the source
a. P only
b. Q only
c. Both P and Q
d. Neither P nor Q
14. Let πA be a problem that belongs to the class NP. Then which one of the following is TRUE?
a. There is no polynomial time algorithm for πA
b. If πA can be solved deterministically in polynomial time, then P=NP
c. If πA is NP-hard, then it is NP-complete
d. πA may be undecidable
15. Which one of the following languages over the alphabet {0,1} is described by the regular expression: (0+1)*0(0+1)*0(0+1)*?
a. The set of all strings containing the substring 00.
b. The set of all strings containing at most two 0’s.
c. The set of all strings containing at least two 0’s.
d. The set of all strings that begin and end with either 0 or 1.
16. Which one of the following is false?
a. There is a unique minimal DFA for every regular language.
b. Every NFA can be converted to an equivalent PDA.
c. Complement of every context-free language is recursive.
d. Every non deterministic PDA can be converted to an equivalent deterministic PDA.
17. Match all items in Group 1 with correct options from those given in Group 2
Group 1 Group 2
P. Regular expression 1. Syntax analysis
Q. Pushdown automata 2. Code generation
R. Dataflow analysis 3. Lexical analysis
S. Register allocation 4. Code optimization
a. P – 4, Q – 1, R – 2, S – 3
b. P – 3, Q – 1, R – 4, S – 2
c. P – 3, Q – 4, R – 1, S – 2
d. P – 2, Q – 1, R – 4, S – 3
18. Consider the program below
#include
int fun(int n, int *f_p)
{
int t,f;
if(n<=1) {
*f_p=1;
return 1;
}
t=fun(n-1,f_p);
f=t+ *f_p;
*f_p=t;
return f;
}
int main() {
int x=15;
printf ("%d", fun(5, &x));
return 0;
}

The value printed is:
a. 6
b. 8
c. 14
d. 15
19. The coupling between different modules of a software is categorized as follows:

I. Content Coupling
II. Common Coupling
III. Control Coupling
IV. Stamp Coupling
V. Data Coupling
Coupling between modules can be ranked in the order of strongest (least desirable) to weakest (most desirable) as follows:
a. I – II – III – IV – V
b. V – IV – III – II – I
c. I – III – V – II – IV
d. IV – II V – III – I
20. Consider the HTML table definition given below:

ab	cd
	ef	gh
ik

The number of rows in each column and the number of columns in each row are:
a. <2,2,3> and <2,3,2>
b. <2,2,3> and <2,2,3>
c. <2,3,2> and <2,3,2>
d. <2,3,2> and <2,2,3>
21. An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?
a. 0.453
b. 0.468
c. 0.485
d. 0.492
22. For the composition table of a cyclic group shown below

* a b c d
------------
a a b c d
b b a d c
c c d b a
d d c a b
Which one of the following choices is correct?
a. a, b are generators
b. b, c are generators
c. c, d are generators
d. d, a are generators
23. Which one of the following is the most appropriate logical formula to represent the statement : “Gold and silver ornaments are precious”. The following notations are used:
G(x):x is a gold ornament
S(x):x is a silver ornament
P(x):x is precious
a. For all x (P(x) -> (G(x) ^ S(x)))
b. For all x ((G(x) ^ S(x)) -> P(x))
c. There exists x ((G(x) ^ S(x)) -> p(x))
d. For all x ((G(x) v S(x)) -> P(x)
24. The binary operation o is defined as follows
P Q P o Q
T T T
T F T
F T F
F F T

a. ¬ Q o ¬ P
b. P o ¬ Q
c. ¬ P o Q
d. ¬ P o ¬ Q
25. ∫0 π/4 ((1-tan x)/(1+tan x)dx ) evaluates to
a. 0
b. 1
c. ln 2
d. ½ ln 2
26. Consider the following well-formed formula:
I. not for all x (P(x))
II. not there exists x (P(x))
III. not there exists x (not P(x))
IV. there exists x(not P(x))
a. I and III
b. I and IV
c. II and III
d. II and IV
27. Given the following state table of FSM with two states A and B, one input and one output:
Present State A Present State B Input Next State A Next State B Output
0 0 0 0 0 1
0 1 0 1 0 0 1
1 0 0 0 1 0
1 1 0 1 0 0
0 0 1 0 1 0
0 1 1 0 0 1
1 0 1 0 1 1
1 1 1 0 0 1
If the initial state is A=0, B=0, what is the minimum length of an input string which will take the machine to the state A=0, B=1 with Output =1?
a. 3
b. 4
c. 5
d. 6
28. Consider a 4-stage pipeline processor. The number of cycles needed by the four instructions I1, I2, I3, I4 in stages S1 ,S2 ,S3 ,S4 is shown below:

S1 S2 S3 S4
I1 2 1 1 1
I2 1 3 2 2
I3 2 1 1 3
I4 1 2 2 2
What is the number of cycles needed to execute the following loop:
For (i=1 to 2) { I1; I2; I3; I4; }
a. 16
b. 23
c. 28
d. 30
29. Consider a 4-way set associative cache (initially empty) with total 16 cache blocks. The main memory consists of 256 blocks and the request for memory blocks is in the following order:
0, 255, 1, 4, 3, 8, 133, 159, 216, 129, 63, 8, 48, 32, 73, 92, 155
Which one of the following memory block will not be in cache if LRU is used?
a. 3
b. 8
c. 129
d. 216
30. Consider a system with 4 types of resources R1 (3 units). R2 (2 units), R3 (3 units), R4 (2 units). A non-preemptive resource allocation policy is used. At any given instance, a request is not entertained if it cannot be completely satisfied. Three processes P1, P2, P3 request the resources as follows if executed independently.
Process P1:
T=0: requests 2 units of R2
T=1: requests 1 unit of R3
T=3: requests 2 units of R1
T=5: releases 1 unit of R2 and 1 unit of R1
T=7: requests 1 unit of R3
T=8: requests 2 units of R4
T=10: finishes
Process P2:
T=0: requests 2 units of R3
T=2: requests 1 unit of R4
T=4: requests 1 unit of R1
T=6: releases 1 unit of R3
T=8: finishes
Process P3:
T=0: requests 1 unit of R4
T=2: requests 2 units of R1
T=5: releases 2 units of R1
T=7: requests 1 unit of R2
T=8: requests 1 unit of R3
T=9: finishes
Which one of the following statements is true if all 3 processes run concurrently starting at time t=0?
a. All processes will finish without any deadlock
b. Only P1 and P2 will be in deadlock
c. Only P1 and P3 will be in deadlock
d. All 3 processes will be in deadlock
31. Consider a disk system with 100 cylinders. The request to access the cylinders occur in following sequence:
4, 34, 10, 7, 19, 73, 2, 15, 6, 20
Assuming that the head is currently at cylinder 50, what is the time taken to satisfy all requests if it takes 1 ms to move from one cylinder to adjacent one and shortest seek time first policy is used?
a. 95ms
b. 119ms
c. 233ms
d. 276ms
32. In the following process transition diagram for a uniprocessor system, assume that there are always some processes in the ready state:
Now consider:

I. If a process makes transitions D, it would result in another process making transition A immediately
II. A process P2 in blocked state can make transition E while another process P1 is in running state
III. The OS uses preemptive scheduling
IV. The OS uses non-preemptive scheduling
Which of the above are TRUE?
a. I and II
b. I and III
c. II and III
d. II and IV