Expert: Josh - 3/27/2006

Question
Hi Josh, i hope you can help with the following i can not work it out: There are eight doors in a row. Behind each door is a number. Each number is less than twenty. No two numbers are the same, and the numbers on the doors are in order.

- Three of the numbers are prime numbers.
- Two of the numbers are square numbers.
- Two of the numbers are multiples of five.
- Five of the numbers are even.
- Three is a factor of two of the numbers.
- There are a pair of consecutive numbers and four consecutive numbers on the doors.
- The numbers sum to 93.

Whats the numbers on the doors?

Hope to hear and my thanks in anticipation.

Kind regards,

Ben

Answer
Hi Ben,

Let's summarize this information.
Let n(1),n(2),...,n(8) represent the numbers hidden behind each door.

Fact 1: "All numbers less than twenty"
n(i)<20 for 1<=i<=8

Fact 2: "No two numbers are the same"
n(i) NOT EQUAL TO n(j) for i NOT EQUAL TO j; 1<=i,j<=8

Fact 3: Three numbers are prime.
Let n(j),n(k),n(l) assume a value from the set of prime numbers between 1 and 20, P={2,3,5,7,11,13,17}

Fact 4: Two numbers are square.
Let n(a),n(b) be the square numbers, taken from the set S={2^2,3^2,4^2}={4,9,16}. These are the only feasible ones in the range 1<=n<=20
[Note: "a" and "b" may take values of 1,2,3,4,5,6,7,8; but being different from "j", "k" and "l" (since squared numbers can never be prime). These indices simply label which door.]

Fact 5: Two numbers are multiples of five.
Again, let n(c),n(d) be the numbers concerned, taking values from the set, F={5,10,15}

Fact 6: Five numbers are even.
Let n(e),n(f),n(g),n(h) and n(i) be the even numbers, taken from E={2,4,6,8,10,12,14,16,18}

Fact 7: Three is a factor for two of the numbers.

I cannot think of a quick and elegant way of solving this other than by trial-and-error.

On paper, I have written the following using a blue pen.

(x3) <- P={2,3,5,7,11,13,17}
(x2) <- S={4,9,16}
(x2) <- F={5,10,15}
(x5) <- E={2,4,6,8,10,12,14,16,18}

Interpretations:
* 3 numbers are to be selected from the set of prime numbers, P.
* 2 numbers are to be selected from the set of squared numbers, S.
* 2 numbers are to be selected from set F, multiples of five.
* 5 numbers are to be selected from set E, the even integers.

Personal observations:
Because we are picking 8 numbers in total, 3 of these must come from set P, while 5 must come from set E, there will be substantial overlapping. i.e., certain numbers will satisfy one or more of the properties and this is what we should look for.

I suggest using a pencil to circle each number that you pick and erase the mark if it turns out to be inappropriate.

Another comment is that the prime numbers in P are mutual exclusive with most of the elements in set S, F and E; with the exception of "5". It is hard to find a number which appears in the union of P and S. I cannot find any common number between P and F, likewise for P and E.

However, notice that "4" appears in S and E, "5" appears in both P and F. Similarly, "10" appears in F and E and "16" appears in both S and E. So, there is a good chance that some of these should be included. This gets us going.

So far, we have
(2 more numbers to choose) <- P={2,3,"5",7,11,13,17}
(no more numbers to choose) <- S={"4",9,"16"}
(no more numbers to choose) <- F={"5",10,15}
(2 more numbers to choose) <- E={2,"4",6,8,"10",12,14,"16",18}

"." denotes a selected number

After selecting four numbers, we have 4 more integers to pick. Two must come from set P, another two from set E.

The intermediate sum is "4+5+10+16"=35.
We still have to make up the difference 93-35=58.

On average, the remaining four numbers have magnitude 58/4~=14.5. Therefore, start focusing on larger numbers and the remaining conditions.

With the exception of the "four consecutive numbers on the doors," I think most of the other conditions are met by selecting "11", "12", "17" and "18".

I hope this will help you get closer to finding the answer.