# Make $2471$ out of first seven powers of two

Can you make $$2471$$ using first seven powers of two: $$1,2,4,8,16,32,64$$, each exactly once?

Only the basic operations are allowed: $$+,-,\times,\div$$, and parentheses $$(\space)$$.

Concatenation, exponentiation,... or any other trickery is not allowed.

Hint:

If parentheses were not allowed, it would be impossible to make it.

Similar Example:

Similarly, $$595$$ can be made with first six powers of two and same rules. Example Solution:

$$595 = (1 + 4/8 + 16)\times(32 + 2)$$ is unique solution.

Here's what I found (after way too much fiddling around):

$$\mathbf{(8+32)\times(64-2-\frac{4}{16})+1 = 2471}$$

Most of my useless fiddling was caused by this super annoying red herring:

$$2471$$ in binary is $$100110100111$$. That's very promisingly $$100110$$ repeated, plus 1, which translates into decimal as $$38\times65+1$$.

Alternately, it could be $$100111$$ repeated, minus 64, (that is $$39\times65-64$$), but even though there are dozens of ways to construct both these approaches (while using only six or seven powers of two), none of them will actually work because of one duplicated number or another.

Finally, I reached the solution by starting over with my initial approach, which was to

greedily go as near the goal as possible, first with two numbers: $$64\times32 = 2048$$, then with three: $$64\times(32+8) = (64+16)\times32 = 2560$$, and then work out a way to make the remaining difference disappear. On the first time around, I had convinced myself that this wasn't going to work while sticking to integers, which was why I took the detour to binary at that point. (Bye-bye, most-of-an-hour.)

• Congrats! This should be the unique solution If I haven't missed anything. Aug 23, 2019 at 19:56
• I used the "greedy method" to try to solve it too, but that got me absolutely nowhere. (Er, somewhere, but I quickly noticed what I was doing wasn't going to work. When I realized the amount of possibilities I gave up.) Aug 26, 2019 at 4:07