Which six-figure number, all of whose digits are different, has the property that when it is multiplied by any of the integers 1,2,3,4,5 or 6, the resultant product is another six-figure number containing the same set.
Background of "Pecking Order" https://modernfarmer.com/2016/03/pecking-order/ - In a flock of chickens, there is a hierarchy established through a pecking order. Each pair of chickens determine, who is dominant amongst them.
How can you represent "pecking order" amongst 3 chickens? What are the different possible outcomes?
What does it mean for the pecking order to be transitive? What kind of pecking orders are possible with transitive property.
In our scheme of things, pecking orders need not be transitive. So amongst any pair, any of the two chickens can be dominant. In such a scenario, we want to come with some definition of "King Chicken".
Ask the kids to propose definitions of king chicken. (at least 3 definitions)
The chicken who can peck most number of chickens.
Since there might not be a chicken that a "King" pecks, what might be the next best option - guide towards having a Field Marshall. We will use this definition because it leads to interesting results.
Is it possible that with our definition, a King chicken does not have the highest peck count? In the extreme case, can the chicken with lowest peck count be a king? Construct such an example.
Similarly, can you construct a case where the chicken with highest peck count is not the king? (Can not be done) - Can you prove that this can't be done.
Answer: Suppose C has highest peck count but is not a king. There must be some C' which C can't peck directly or through its field marshals. Thus C' must peck C and all of C's field marshals, thereby having higher peck count than C. Proof by contradiction.
Rest of the class is dedicated around the question - "How many kings can a flock of n chickens have"
Ask students for their theoretical bounds
Is it possible to have zero kings? (Think about what we just proved)
What are the other possibilities of number of kings
Let students work with flock of size 1, 2, 3, 4, 5 and so on to figure out the possible number of kings in the flock. Have them make conjectures and prove them if possible.
Capture the conjectures and proofs. Guiding questions:
How can we arrange a flock of any size to have exactly one king?
A chicken by itself in a barnyard is obviously a king. What happens if there are two chickens? What are the possibilities with three chickens - how many of the chickens are kings in each case?
Find a way for a flock of four chickens to have exactly one or three kings. Can you show that it is impossible for such a flock to have 2 or 4 kings?
Suppose we have a flock of n chickens with k kings. Show that there exists a flock of n+1 chickens with exactly k kings.
Suppose we have a flock of n chickens where every chicken is a king. Show that you can construct a flock of n+2 chickens so that every chicken is a king.
Introduce two more chickens C1 and C2, so that every chicken in the old flock pecks C1, C1 pecks C2, and C2 pecks every other chicken in the old flock. Now show that every chicken in old flock continues to be a king, and C1 and C2 are also kings
Construct a pecking order for a flock with an odd number of birds where every chicken is a king
Arrange chickens in a circle and have every chicken peck the next (n-1)/2 chickens
For an even size flock, can we show that not everyone can be a king?
Note that this is true for size 2 and 4 flocks. However, its not true for 6 onwards!
c1->c2->c3->c4->c5->c6->c7 and c1->c3->c5->c1 and c2->c6->c4->c2
What does this imply about the possibility of all kings in a flock of any size?
Possible for every size except 2 and 4
Can a flock have exactly two kings?
Let students try and figure it out, with different size flock. Let students form a hypothesis.
Lets think about another problem. If any chicken C is pecked by other chickens, then one of the other chickens must be a king. Illustrate this on the five chicken problem. Let students think about this and try to reason it out.
It is clear that C must peck every other chicken apart from the set that peck it. Thus the king of the set must also be an overall king.
Now can we think about exactly two kings?
If C1 and C2 are the only two kings, with C1 pecking C2, then given the previous lemma, C1 can't be pecked by anyone, which means that C2 can't be the king.
What can you now say about a flock with only one king
That king must directly peck every other chicken
Homework:
(536Dudeney - 290) The outside wheels of a car, running on a circular track, are going twice as fast as the inside ones. What is the diameter of the circle traced by outer wheels? The wheels are five feet apart on the two axles.