Can the magician guess the number?

Question

In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers $(acb)$, $(bca)$, $(bac)$, $(cab)$, and $(cba)$, to add these five numbers, and to reveal their sum, $N$. If told the value of $N$, the magician can identify the original number, $(abc)$.

This is an excerpt from the 1986 AIME (problem 10), in which $N = 3194$. But can the magician actually guess the number, for any $N$? How?

(I'd prefer an answer that allows for $a = 0$, i.e. replace "three digit number" with any number between $0$ and $999$. But the problem as stated, with $a \ge 1$, is fine too.)

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One idea:

We can prove that the original 3-digit number is determined modulo $666$. To see this, if $m = 100a + 10b + c$ is the original number, then $N = 222(a+b+c) - m$. Modulo $222$, this gives us $m \equiv -N$. And modulo $9$, $m \equiv a+b+c$, therefore $N \equiv 222m - m$, and solving we get $m \equiv 2N$. Since $m$ is determined mod $9$ and mod $222$, it is determined mod $\text{lcm}(9,222) = 666$. Specifically, $m \equiv 443N \pmod{666}$.

But this doesn't quite solve the problem, because $666 < 1000$.

Answer 1

With the calculated value of $m\bmod666$ we evaluate possible values of $m=\overline{abc}$. If there is only one possible number ($m>333$) then we are done. Otherwise there will be two candidates $m_1=m$ and $m_2=m+666$ and we can check them easily to see whether they produce $N$ exactly.

$m_1$ and $m_2$ must produce different values of $N$. To see why, consider the addition $m_1+666$ and four cases depending on whether there is a carry-over from the units and tens places. For $N$ to remain unchanged we must have $222(\Delta a+\Delta b+\Delta c)-666=0$, or $S=\Delta a+\Delta b+\Delta c=3$.

• No carry: $\Delta a=\Delta b=\Delta c=6$. $S=18$.
• Carry from units place: $\Delta c=-4, \Delta b=7, \Delta a=6$. $S=9$.
• Carry from tens place: $\Delta c=6, \Delta b=-4, \Delta a=7$. $S=9$.
• Carry from both units and tens: $\Delta c=-4, \Delta b=-3, \Delta a=7$. $S=0$.

Since $S$ is not 3 in all cases, $N$ must change in going from $m_1$ to $m_2$. Hence we will always get a unique, correct value for $m$ from $m_1$, $m_2$ and $N$.

Answer 2

$acb + bca + bac+cab + cba = 222(a+b+c) - abc$

This number is unique to $a,b,c$ because:

Suppose $222(a + b + c) - abc = 222(d+e+f) - def$ and wolog $abc \ge def$.

then $abc - def = 222([a+b+c] - [d+e+f])$. As $abc - def < 1000$, $[a+b+c]-[d+e+f] = k$ where $k = 0,1,2,3,4$.

$abc = 100a + 10b + c \equiv a + b + c \mod 9$

and

$def \equiv d+e+f \mod 9$

So $222k = abc - def \equiv [a+b+c]-[d+e+f] = k \mod 9$.

And therefore $221k \equiv 0 \mod 9$

so $5k \equiv 0 \mod 9$.

Of $k = 0,1,2,3,4$, $k = 0$ is the only possibility.

So $k = (a+b+c) - (d+e+f) = 0$

and $abc - def = 222*0 =0$ so $abc - def$.

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FWIW I figured out how a magician can do this in his head quickly.

Take the result M. Iterate and add the digits to get a single digit. Double that and iterate to get a single digit $k$. Multiply by $2$ and then by $111$ -- (if $2k = 1b$ this is just $1.(b+1).(b+1).b$-- Easy to do in your head) --- to get $J = 222k$.

If $J < M$ add $1998$ (add $2000$ and subtract $2$) until you get $J \ge M$.

Then the original number is $J - M$.

Example: Result is $3194$. Add the digits to get $8 \mod 9$. Double to get $16 \equiv 7 \mod 9$. Multiply by $2$ to get $14$ and multiple by $111$ to get $1554$. Add $2000 - 2$ so get $3552$. Subtract $3194$ to get $358$ which is the original number.