# How can my friend take the item for only $3,000? I have an item that I would like to give to my friend. Its value is 3,000 dollars, and this base value cannot be changed. My friend would like to buy it from me for exactly 3,000 dollars, and I'd love to give it to him, but there is one problem: The government has (strangely) created a rule that states that one must sell an item for at least two times its value but no more than ten times its value. That means I can only sell that item to my friend from 6-30 thousand dollars. But we've devised a plan: We've decided that we will try and sell the item back and forth to each other to try to get a manipulation. So that the end result will be that I earned 3,000 dollars and my friend gained the item. Unfortunately, the government is on our tail: We can only trade 3 times (each time that item travels is one time, back and forth is 2 times). Once we do more than that, the government will see what we are trying to do and revoke the item entirely! What and how can we trade so that the net result ends up with me and my friend swapping the item for 3,000 dollars? I have a picture below (warning, I am artistically challenged) that depicts what the trades would look like. The tomato in the picture represents an item, and the "$" means that money is being sent.

Notes and hints:

There is a logical answer to this riddle, and there is no wordplay. I will hand out additional hints if necessary. Good luck!

• The riddle has been solved, but if you have a different, unspecified way of solving this, please post it below! – Assafi Cohen-Arazi Nov 30 '17 at 2:10

There are lots of easy ways to do this. For instance:

You sell it to the friend for \$6,000. They sell it back for \$9,000. You sell it for \$6,000 again. Really, any solution to$a+c=b+1$where$a,b,c\in[2,10]$works, and there are a lot of those. Just multiply$a$,$b$, and$c$by \$3000 to get your solution.

• Wow! I never really thought of doing it this way. I'll mark yours as accepted but post an alternative that was the intended answer. Congrats! – Assafi Cohen-Arazi Nov 30 '17 at 2:00
• Wouldn't its value become 6000 after the first transaction? – ibrahim mahrir Nov 30 '17 at 2:02
• @ibrahimmahrir - The question states that its value is 3000 dollars, so I assumed that was fixed. (If it were not, then you could never get that difference in this way - the difference would always be at least the previous value, and the first time through it would be at least double.) – Deusovi Nov 30 '17 at 2:04
• @ibrahimmahrir No, the value is declared by the government as 3,000 dollars fixed (they are very bossy). – Assafi Cohen-Arazi Nov 30 '17 at 2:10
• @AssafiCohen-Arazi Ah I see! Then Deusovi has already handed out the formulae to get all the solutions. – ibrahim mahrir Nov 30 '17 at 2:14

@Deusovi had figured it out logically, but my take on the problem goes like this:

Sell the requested item for 6,000 dollars to the friend, then have them sell you a different item with a base value of 5,000 dollars for $20,000, then give the new item back for 17,000 dollars. • You can't give the new item back for $17,000, it will break the rules as its value is $10,000 so it must be sold for at least $20,000. – ibrahim mahrir Nov 30 '17 at 2:20
• @ibrahimmahrir Good catch! I meant for the base price to be $5,000 – Assafi Cohen-Arazi Nov 30 '17 at 2:20 Branching off of @Deusovi answer I wanted to see how many actual solutions we could find. If we expand of of his solution but allow payments not multiples of \$3,000 we have the equation $a+c =b+3000$ where $a,b,c \in [6000,30000]$
With a quick bit of python there are 220,531,501 solutions to this puzzle. I did not include cents because dear god this was already slow enough.