Prove The Mathematical Fact

Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.

Here is the proof!

Proof of The Mathematical Fact!

What was that fact?

For a moment, let's assume that such group of 2 numbers exists whose product is equal to sum of rest 13 numbers taken out of 15 numbers.

Let x and y be those numbers in group B. Now x and y can be any number from 1 to 15. 

As per condition,

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 - x - y = xy 

120 = xy + x + y 

Adding 1 to both sides,

121 = xy + x + y + 1

121 = x( y + 1 ) + 1( y + 1 )

121 = ( x + 1 ) ( y + 1 ) 

Since x & y are the numbers in between 1 to 15, possible value of x & y satisfying the above equation is 10. But x & y are must be 2 different number. Hence, our assumption goes wrong here!

So, the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B. 

What Could Be The Product?

Zach chooses five numbers from the set {1, 2, 3, 4, 5, 6, 7} and tells their product to Claudia. She finds that this is not enough information to tell whether the sum of Zach’s numbers is even or odd. What is the product that Zach tells Claudia?

Here is the way to get that product!

Guessing The Correct Product in Question!

What was the question?

When Zach tells the product of 5 numbers that he has chosen he indirectly conveying product of 2 un chosen numbers.

For example, product of all numbers in set = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 and if Zach tell product 1 x 2 x 4 x 5 x 7 = 280 then obviously the product of numbers that he hasn't chosen is 5040/280 = 18 = 3 x 6.

Now, there are only 2 products viz. 12 and 6 which have more than 1 pair of numbers. 

The product 12 can be from pairs - (3,4) and (6,2)

The product 6 can be from pairs - (1,6) and (2,3) 

Here if the sum of un chosen numbers is odd (or even) then sum of other 5 chosen numbers also must be odd (or even).

In above cases, 6 has pairs whose sum is odd always and hence sum of other 5 numbers would be odd. In that case, Claudia would have been sure with if sum of numbers selected by Zach is either odd or even.

While in other case, the product 12 has 1 pair having sum odd (3,4) and other pair having sum even (6,2). Hence, Zach must have 'indirectly' suggested product 12 as a product of un chosen numbers that's why Claudia is saying that she doesn't know if the sum of numbers selected by Zach is even or odd.

Hence, the product of numbers selected by Zach = 5040 / 12 = 420.    

 

Difficult Puzzle of Sum and Product

Sum Sam and Product Pete are in class when their teacher gives Sam the Sum of two numbers and Pete the product of the same two numbers (these numbers are greater than or equal to 2). They must figure out the two numbers.

Sam: I don’t know what the numbers are Pete.

Pete: I knew you didn’t know the numbers… But neither do I.

Sam: In that case, I do know the numbers.

What are the numbers?

Want to know those numbers?

Solution - Difficult Puzzle of Sum and Product

Wait, read the puzzle once! 

Let's remind that the numbers are greater than or equal to 2; means those can't be either 0 or 1.

Now take a look at what Sam & Pete says - 

Sam: I don’t know what the numbers are Pete.

Pete: I knew you didn’t know the numbers… But neither do I.

Sam: In that case, I do know the numbers.

If Sam was told 4 then straightway he would have numbers 2,2 in mind as 3,1 combination is invalid.

If teacher had told Sam 5 as a sum then too Sam had correct pair of numbers 2,3 immediately as 4,1 or 5,0 are invalid combinations. 

So Sam must have at least number 6. Valid combinations for this sum are (2,4), (3,3).

If it was (3,3) then Pete would had 9 & he would have identified this combination correctly as (9,1) is not valid combination. Since he too didn't know exact numbers, it must be some different combination.

And if teacher had told Pete 8 then too he would have easily figured out correct combination of (2,4) as (8,1) is not valid.

So Pete can't have number 1,2,3,4,5,6,7, 8 or 9 or 11.

Now if he had 10 then only possible combination (2,5) and he would have that immediately. So he wouldn't have made the statement that he too didn't know numbers.

Let's assume that he had number 12 as product. Now in this case valid combinations are (2,6), (3,4).  The sums of these valid combinations are 8 & 7 respectively.

Now depending on what sum the Sam had; he can identify the correct pair of numbers easily.

The Ages of Three Daughters

I was visiting a friend one evening and remembered that he had three daughters. I asked him how old they were. “The product of their ages is 72,” he answered. Quizzically, I asked, “Is there anything else you can tell me?” “Yes,” he replied, “the sum of their ages is equal to the number of my house.” I stepped outside to see what the house number was. Upon returning inside, I said to my host, “I’m sorry, but I still can’t figure out their ages.” He responded apologetically, “I’m sorry, I forgot to mention that my oldest daughter likes strawberry shortcake.” With this information, I was able to determine all three of their ages. How old is each daughter?

How old they are? Click here! 

Source 

Calculation of The Ages of Three Daughters

How it was asked to find with some clues? 

The product of their age is 72; so possible combinations are,

3 x 3 x 8 = 72                       3 + 3 + 8 = 14

2 x 6 x 6 = 72                       2 + 6 + 6 = 14

2 x 2 x 18 = 72                     2 + 2 + 18 = 22

2 x 3 x 12 = 72                     2 + 3 + 12 = 17

2 x 4 x 9 = 72                       2 + 4 + 9 = 15

3 x 4 x 6 = 72                       3 + 4 + 6 = 13
 
If any of last 4 combinations was correct then I would have come to know ages of daughter straightaway as their sum is unique. So friend's house number as he must have told must be 14. That's why I was uncertain with the correct combination as they are appearing twice.

In next clue, he indirectly suggest he has only 1 oldest daughter. So 2,6,6 is eliminated as in that case he would have 2 oldest daughters.

So the correct combination is 3,3,8. That means he had oldest daughter of age 8 & younger twins.