Henry Ernest Dudeney/Puzzles and Curious Problems/97 - Letter Multiplication/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $97$

Letter Multiplication
In this little multiplication sum the five letters represent $5$ different digits.
What are the actual figures?
There is no $0$.
    S E A M
x         T
-----------
  M E A T S


Solution

    4 9 7 3
x         8
-----------
  3 9 7 8 4

Proof

Note that:

$M < T$, because $MEATS = SEAM \times T < 10000 \times T = T0000$.
$M < S$, because $MEATS = SEAM \times T < SEAM \times 10 = SEAM0$, and $M \ne S$.

Also $S \ne T$.

By observing the lower triangular part of the modulo $10$ multiplication table below:

$\begin{array}{c|cccccccc} T \backslash M & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 2 & 2 \\ 3 & 3 & \color {red} 6 \\ 4 & 4 & \color {red} 8 & 2 \\ 5 & 5 & 0 & 5 & 0 \\ 6 & 6 & 2 & \color {red} 8 & 4 & 0 \\ 7 & 7 & \color {red} 4 & 1 & \color {red} 8 & 5 & 2 \\ 8 & 8 & \color {red} 6 & \color {red} 4 & 2 & 0 & 8 & 6 \\ 9 & 9 & \color {red} 8 & \color {red} 7 & \color {red} 6 & 5 & 4 & 3 & 2 & \\ \end{array}$

we see that only the $\color { red } {\text {red} }$ numbers satisfy the conditions for $S$.


Now we consider an additional condition:

$M \ge \floor {\dfrac {S \times T} {10}}$

i.e. the tens digit of $S \times T$ cannot exceed $M$.

Otherwise $\floor {\dfrac {S \times T} {10}} \ge M + 1$,

and $SEAM \times T > S000 \times T = 10000 \times \dfrac {S \times T} {10} \ge 10000 \paren {M + 1} > MEATS$.


Hence only the $\color { red } {\text {red} }$ numbers in:

$\begin{array}{c|ccc} T \backslash M & 2 & 3 & 4\\ \hline 3 & \color {red} 6 \\ 4 & 8 & 2 \\ 5 & 0 & 5 & 0 \\ 6 & 2 & 8 & 4 \\ 7 & \color {red} 4 & 1 & 8 \\ 8 & 6 & \color {red} 4 & 2 \\ 9 & 8 & 7 & 6 \\ \end{array}$

remain.


Using only the values of $M, T, S$, we can determine the value of $SEAM$ by the formula:

$SEAM = \dfrac {S00M0 - M00TS} {10 - T}$

since:

\(\ds SEAM \times \paren {10 - T}\) \(=\) \(\ds SEAM0 - MEATS\)
\(\ds \) \(=\) \(\ds 10^4 \paren {S - M} + 10^3 \paren {E - E} + 10^2 \paren {A - A} + 10 \paren {M - T} + \paren {0 - S}\)
\(\ds \) \(=\) \(\ds 10^4 S + 10 M - 10^4 M - 10 T - S\)
\(\ds \) \(=\) \(\ds S00M0 - M00TS\)

Now we inspect the $3$ remaining cases using the formula.

$M = 2, T = 3, S = 6$ gives:

$6EA2 = \dfrac {60020 - 20036} {10 - 3} = 5712$

$M = 2, T = 7, S = 4$ gives:

$4EA2 = \dfrac {40020 - 20074} {10 - 7} = 6648 \tfrac 2 3$

$M = 3, T = 8, S = 4$ gives:

$4EA3 = \dfrac {40030 - 30084} {10 - 8} = 4973$


Only $4973$ matches the required form of $SEAM$.

Hence the solution given is the unique solution.

$\blacksquare$


Sources

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