# Alphametic puzzle for year 2017

In east Asia 2017 is the year of a rooster.

$$ABCD=\left(A \times CA^A + \frac{C}{D} \right) \times D$$ • So what's the puzzle?
– bleh
Jan 6, 2017 at 2:16
• Did you find that image and puzzle elsewhere? If so, please cite your source. (I just wondered because of the text at the bottom of the image.) Jan 6, 2017 at 2:21
• @bleh This is an alphametic puzzle. Find each letter (or rooster). Jan 6, 2017 at 2:27
• @P.-S.Park But like put that in the puzzle?
– bleh
Jan 6, 2017 at 2:29
• @randal'thor I made the image. But the weathercock icon was made by Tintins. I inserted the text by the copyright policy of the site. Jan 6, 2017 at 2:29

The solution is, of course,

A = 2, B = 0, C = 1, D = 7.

We can prove that this is a unique solution like so:

Looking at A,

A is the first digit of a 4-digit number, and so cannot be 0. A cannot be greater than 3 either, since 144 is already 5 digits. A cannot be 3, since the only possible value for C keeping the result under 4 digits would be 1, but then there would be no solution for D. A cannot be 1, since then we would have (CD+C) equaling a 4-digit number, which is impossible. Therefore A must be 2.

Looking at C,

C is the first digit of a 2-digit number, and so cannot be 0. C cannot be greater than 3 either, since 2*422 is 3528, whose first digit is greater than 2. C cannot be 2, since A is 2. C cannot be 3, since 2*322 is 2048, and to keep the first digit 2, D is forced to be 1. However, subbing these values for A, C, and D in the right-hand side, the result is 2051, which doesn't fit the left-hand side. Therefore C must be 1.

Looking at D,

2*122 is 288. To get a 4-digit number starting with 2, D must be greater than or equal to 7. However, D cannot be 8 or 9 since the right-hand side would be 2305 or 2592 respectively, neither of which matches the left-hand side. D = 7 results in 2017, which fits and forces B to be 0.

With the question, as it is now.
This could be accepted

A,B,C = 0
D = 1

• ABCD does not mean A×B×C×D Jan 6, 2017 at 3:03
• Yes it does. ABCD without the concatenation sign on top is A*B*C*D
– bleh
Jan 6, 2017 at 3:10
• not in this context. Jan 10, 2017 at 22:28

I don't know if there are several answers, but here is one possibility:

C is equal to 0, A and B are equal to any existing number, and D is equal to any number different of 0.

• ABCD does not mean A×B×C×D Jan 6, 2017 at 3:03