Odd and even coins

Question

Two friends are playing a coin game. You need to give to your friend, one coin that has an even value and another coin that has an odd value. For example, a dime would be the even value since its value is $10$ and a nickel could be the odd value since its value is $5$. Now your friend is going to take the two coins and put one in each hand. He or she must follow the below:

1) Triple the value of the coin in the right hand 2) Double the value of the coin in the left hand

Then add the two results together. If the sum is even, the even coin aka the "dime" is in the right hand. If the sum is odd, the even coin aka the "dime" is in the left hand.

Why do this work?

I actually did this problem but I cant seem to find the reason why this works. I am guessing it has to do with the fact that there is an odd value and even value involved. Any ideas?

Also, I would love to know if this works for other coins that might be odd or even as well like quarter or penny.

Answer 1

Say the value of the even coin is $2m$, and the value of the odd coin is $2n+1$. If the even coin is in the right hand, the calculation is

$$3(2m)+2(2n+1)=6m+4n+2=2(3m+2n+1)\;,$$

which is even. If the even coin is in the left hand, the calculation is

$$3(2n+1)+2(2m)=6n+4m+3=2(3n+2m+1)+1\;,$$

which is odd.

Answer 2

This works because whenever you double a number, even or odd, you get an even number. When you add an even number to any number, its oddness or evenness doesn't change. Therefore, the number that gets doubled does not affect the outcome. The number that gets tripled is the number that determines the answer. Three odds give an odd and three evens give an even. So if the answer is odd, the odd coin is in the right hand and if the answer is even, the even coin is in the right hand.

Answer 3

The dime is a value $10$. Thus multiplied by any integer, in your case tripled or doubled, would yield an even number. So in whichever hand it is, it will be even.

The nickel is a value $5$. Thus multiplied by an even integer, it would yield an even number. Multiplied by an odd integer, it would yield an odd number. Therefore if it contributes an even or odd value at the end depends in which hand it is.

If the nickel is in the left hand, it gets doubled, therefore the value is even. Then the dime in the right hand is tripled, therefore even. Even value plus an even value is an even value.

If the nickel is in the right hand, it gets tripled, therefore value is odd. Then the dime is in the left hand and is tripled, again even. Even value plus an odd value is an odd value.

Therefore we can see that it is the fact that the nickel is an odd integer that really plays the key factor here. The dime will always contribute an even integer to the final sum, because any integer multiplied by an even integer is an even integer. However depending on in which hand the nickel is, it will either contribute an even value if it is doubled, and an odd value if it is tripled.