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Problematics | Ageing superheroes
Either Black Widow or Captain Marvel or Scarlet Witch is lying. Which superhero is how old?
Fiction is flexible when it comes to the ages of evergreen characters. James Bond has always been fortyish from the 1960s to 2010s, remaining the same age during the Cold War era and in the 21st century. Black Widow was born in 1928 in Marvel comics and in 1984 in the Marvel Cinematic Universe.
That should give Problematics, too, the licence to be flexible with Marvel superheroes. The following puzzle comes with a statutory disclaimer: the ages mentioned are imagined, and any similarity with established fictional ages is coincidental.
#Puzzle 189.1
A puzzler meets the Marvel superheroes Black Widow, Captain Marvel and Scarlet Witch and wants to know their ages. The three women decide to entertain themselves with a series of statements.
(1) Black Widow: Scarlet Witch is elder to me
(2) Captain Marvel: I am the eldest among the three of us
(3) Scarlet Witch: The difference between Black Widow's age and mine is 3
(4) Black Widow: Scarlet Witch is not 27
(5) Captain Marvel: All three of us are above 24
(6) Scarlet Witch: If you double Black Widow's age and add 2, you get the sum of Captain Marvel's age and my age
(7) Black Widow: Either Captain Marvel is 25 or Scarlet Witch is 26 or I am 24 — that is to say, exactly one of these three ages is correct
(8) Captain Marvel: Black Widow is the youngest among the three of us
(9) Scarlet Witch: If Black Widow is the youngest among us, then I am not the eldest — but if she is not the youngest, then I am the eldest
Each woman made three statements, as we can see. Later, the puzzler learns that one of the women gave him three false statements while the other two consistently spoke the truth. That knowledge helps him determine all three ages.
From the experience of previous puzzles of this kind, I want to preempt the possibility of any misunderstanding. So please be careful in analysing statements with multiple alternatives. For example, take statement (7) from Black Widow. If she is telling the truth, then exactly one of the ages in that statement is correct. But if she is lying, then there may be two, three or no correct ages among the three being mentioned in that statement.
Again, look at statement (9) in which Scarlet Witch says only one of a pair of conditions is true. If Scarlet Witch is telling the truth, any one of the two conditions (Black Widow youngest, Scarlet Witch eldest) is true, but not both. If both conditions turn out to be simultaneously true or both simultaneously false, then Scarlet Witch is lying.
Which superhero is how old, and which one is lying?
#Puzzle 189.2
Take a traditional clock with 12 at North, 3 at East, 6 at South and 9 at West. The time is a little before 8:20 (am or pm does not matter). Both the hour and minute hands make the same angle with the vertical running through 12 and 6.
What is the exact time?
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 188.1
Dear Kabir,
The painter is the thief, the sculptor is the accomplice and the art collector is innocent.
Let us number the statements: (1) The painter is not the accomplice. (2) The sculptor is not the thief. (3) The art collector is not innocent.
Possibility #1: If we assume that the sculptor is innocent, the other two are liars. One of the two liars will have made statement (2) about the sculptor, meaning that statement (2) is a lie. This would imply that the sculptor is a thief. This contradicts the original assumption. Therefore, the sculptor is not innocent.
Possibility #2: If we assume that the painter is innocent, the other two are liars. One of the two liar will have made statement (1) about the painter, meaning that statement (1) is a lie. This would imply that the painter is the accomplice. This contradicts the original assumption. Therefore, the painter is not innocent.
Possibility # 3: If we assume that the art collector is innocent, the other two are liars. One of the two liars will have made statement (3) about the painter. If statement (3) is a lie, it implies that the art collector is innocent. This is consistent with the assumption.
Therefore, one among the sculptor and the painter is a thief and the other is an accomplice. For the two to have separate roles, Statements (1) and (2) must be either both true or both false. However, at least one of these statements was made by the art collector and is true. Therefore, both these statements must be true. Accordingly, the painter is the thief and the sculptor is the accomplice.
— Professor Anshul Kumar, New Delhi
#Puzzle 188.2
Each person earns a positive integer salary (no zero, no decimals) that is less than ₹1,00,000. The possible salaries are: 1, 2, 3… 99999. We need to distribute 5 million people across these 99999 distinct salaries, and we want to minimise the number of people sharing the same salary. That means we spread people as evenly as possible across all salary values.
Now 5 million divided by 99999 is 50.0005. This means if every salary value had a maximum of 50 recipients, the total recipients would be 99999 × 50 = 4999950. That would leave 5000000 – 4999950 = 50 people still unassigned any salary value. Those extra 50 people have to go into one or more bracket(s), forcing at least 50 salary groups to have 50 + 1 = 51 people. So at least 51 individuals must earn the same salary. There will be 50 sets of 51 such people.
— Vinod Mahajan, New Delhi
The second puzzle was misunderstood by some readers. It was not a condition that the salaries are to be distributed evenly. But repetitions can be restricted to a minimum only if the distribution is spread out.
There also appears to have been a misunderstanding about what “minimum” means. If two different amounts are to be divided among five people, the split can be 5-0 (all five getting the same amount), 4-1 or 3-2. The minimum number of repetitions is achieved by the third option, with 3 people receiving one salary and 2 people receiving the other salary. This means the minimum number of repetitions is 3 (not 2). We can divide 2 salaries among 5 people with 3 repetitions. We cannot do that if we limit ourselves to 2 repetitions (because there must be 3 repetitions of the other salary).
Solved both puzzles: Professor Anshul Kumar (Delhi), Vinod Mahajan (Delhi), Yadvendra Somra (Sonipat), Sabornee Jana (Mumbai), Ajay Ashok (Delhi)
Solved #Puzzle 188.1: Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Amarpreet (Delhi). Dr Vivek Jain (Baroda), Shishir Gupta (Indore)
Problematics will be back next week. Please send in your replies by Friday noon to problematics@hindustantimes.com
