Can You Solve The 3 Consecutive Numbers Puzzle?

I have found three consecutive integers such that their product is equal to their sum. What could my numbers be? Solve for all possibilities. Then watch the video for a solution. I thank Ken Edwards for creating the problem and Stephen for sending me the solution.

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[funnynotfunny]

Some of his questions can be solved by 10th graders (like this one) while others can't be solved by even PhD students.

[LukeEdward]

Thank you. I am going to start tutoring a 7th grader on Tuesday and this is the PERFECT icebreaker problem.

[dagordon]

Nice work, Ken Edwards. I used x, x+1, & x+2 to get x3+ 3x2- x-3=0 and found a factor of x+3.

[lammy]

I happened to think of all 3 possibilities before even busting out the math, then when I did the math I realized that those were the only 3 solutions.

[NukeMarine]

Did it in my head the bulk force way (not much force required though). Feel silly not thinking of going the formula route as that's cooler. It's a nice puzzle as I would think many don't use negative numbers in the possible solutions to many puzzles.

[rastkolazarevic]

This equation has much more solutions:
For example:
15, 0, -15
15•(-15)•0=15+(-15)+0
0=0 and it is correct
It means
One of them(x, y, z) have to be 0
And the other two have to be opposite numbers.
And finally we have infinite solutions:
(x, y, z)=(n, 0, -n) and of course (1, 2, 3) and (-1, -2, -3)

[Wezleigh]

The only one I didn’t immediately get was -1 0 and 1

[therealincognito]

This was so easy! The first one I found was 1, 2, and 3

[Kiriba]

You solved it the smart way. I instantly know x=0 since 1-1 cancel out and 2 from plugging and chugging. Then I got -2 since every answer would have a negative counter part. I knew there were no other answers since the gap from the answer would increased exponentially.

[fizixx]

Eady....I liked this one. No 'tricks' needed, just straightforward. Thanks for the video.

[VENOM-txgp]

I just read the question and immediately said 1, 2 and 3 .that was easy but for other that must be 0 and a negative group.
This was a lot easier that other videos of him.

[FederationStarShip]

I got this one! I used a similar method but I set x as the smallest of the three so my cubic equation was a bit more complicated

[TheJoaovascorodrigue]

It's the first time I solved one of your problems!

[therealEmpyre]

I am happy to say that I did figure it out, not not with algebra. I realized that the numbers had to be small because the product of three big numbers will be much bigger than their sum. I tried the smallest set of consecutive positive integers (1, 2, 3), and it worked. Then i tried the next set (2, 3, 4) and the product is already bigger then the sum, so I knew there woudn't be any more solutions. then I realized that he said integers, not natural numbers, and it dawned on me that the negatives of 1, 2, and 3 would work too. I thought I was done, thinking that the set couldn't include 0 because the product would just be 0, but then it occurred to me that adding -1 and 1 would cancel out, so -1, 0, and 1 is a third solution.
I know that Presh's algebraic approach is better because not all puzzles can simply be figured out like I did with this one.

[amelia-]

I tried 1, 2, 3 in my mind and it worked .. I didn’t think it would be so easy but there should be other solutions as well

[L]

This was a pretty simple problem. Figured out all 3 solutions relatively quickly.

[jeffreyblack]

You can find it easier than that.
You note the sum is 3*x, while the product has x as a factor. This gives x=0 as an option, and allows you to divide both sides by x, giving (x+1)*(x-1)=3 to find the rest which simplifies to x^2=4.

[maisiebutler]

I'm so stupid I got -1 0 1 but I don't ben think about 123

[carloscantu]

Finally one i got. Ive been watching these videos for an hour

[federicovolpe]

x+x+1+x+2=x(x+1)(x+2)
3x+3=x(x^2+3x+2)
3x+3=x^3+3x^2+2x
x^3+3x^2-x-3=0
Now "just" solve for x
x=-3
x=1
x=-1
In fact, the solutions are
-3;-2;-1
-1;0;1
1;2;3
That's a pretty boring way of solving it but that's the only one I though of, except just random trying (but that doesn't demonstrate there are only 3 solutions)
EDIT: the way of solving that you show in the video is actually more clever, chapeau!