We establish some parameters for the number $x$ such that the equation $x + 2 x = 3 x$ is pandigital. Firstly, among the three numbers, there cannot be repeating digits or $0$. If $5 \divides x$: :$10 \divides 2 x$ and $2 x$ ends in $0$. By Divisibility by 5, $x$ cannot end in $5$. Secondly, $x$ must be divisible by $3$. To see this, consider the concatenation of the three numbers, $\sqbrk {x || 2 x || 3 x}$. Since the digits of this number is a permutation of $\set {1, \dots, 9}$, they sum to $45$. Since $9 \divides 45$, by Divisibility by 9: :$9 \divides \sqbrk {x || 2 x || 3 x}$ We have: {{begin-eqn}} {{eqn | l = \sqbrk {x \vert \vert 2 x \vert \vert 3 x}       | r = 10^6 x + 10^3 \paren {2 x} + 3 x }} {{eqn | o = \equiv       | r = x + 2 x + 3 x       | rr = \pmod 9       | c = Congruence of Powers }} {{eqn | o = \equiv       | r = 6 x       | rr = \pmod 9 }} {{end-eqn}} Hence $9 \divides 6 x$. This gives $3 \divides 2 x$. By Euclid's Lemma: :$3 \divides x$ Finally, as $3 x < 1000$, we must have $100 \le x \le 333$. Putting repeating digits and zeroes into consideration, we list and check all possible candidates from $123$ to $327$: {{begin-eqn}} {{eqn | ll = 12X:       | l = 12X \times 2       | r = 2XX       | c = $2$ is repeated }} {{eqn | ll = 132:       | l = 132 \times 2       | r = 264       | c = $2$ is repeated }} {{eqn | ll = 138:       | l = 138 \times 3       | r = 414       | c = $4$ is repeated }} {{eqn | ll = 147:       | l = 147 \times 2       | r = 294       | c = $4$ is repeated }} {{eqn | ll = 153:       | l = 153 \times 2       | r = 306       | c = $3$ is repeated }} {{eqn | ll = 156:       | l = 156 \times 2       | r = 312       | c = $1$ is repeated }} {{eqn | ll = 159:       | l = 159 \times 2       | r = 318       | c = $1$ is repeated }} {{eqn | ll = 162:       | l = 162 \times 2       | r = 324       | c = $2$ is repeated }} {{eqn | ll = 168:       | l = 168 \times 2       | r = 336       | c = $3, 6$ are repeated }} {{eqn | ll = 174:       | l = 174 \times 2       | r = 348       | c = $4$ is repeated }} {{eqn | ll = 183:       | l = 183 \times 2       | r = 366       | c = $3, 6$ are repeated }} {{eqn | ll = 186:       | l = 186 \times 3       | r = 558       | c = $5, 8$ are repeated }} {{eqn | ll = 189:       | l = 189 \times 2       | r = 378       | c = $8$ is repeated }} {{eqn | ll = 192:       | l = 192 + 384       | r = 576       | c = is the $1$st solution }} {{eqn | ll = 198:       | l = 198 \times 2       | r = 396       | c = $9$ is repeated }} {{eqn | ll = 213:       | l = 213 \times 2       | r = 426       | c = $2$ is repeated }} {{eqn | ll = 216:       | l = 216 \times 2       | r = 432       | c = $2$ is repeated }} {{eqn | ll = 219:       | l = 219 + 438       | r = 657       | c = is the $2$nd solution }} {{eqn | ll = 231:       | l = 231 \times 2       | r = 462       | c = $2$ is repeated }} {{eqn | ll = 234:       | l = 234 \times 2       | r = 468       | c = $4$ is repeated }} {{eqn | ll = 237:       | l = 237 \times 2       | r = 474       | c = $4, 7$ are repeated }} {{eqn | ll = 24X:       | l = 24X \times 2       | r = 4XX       | c = $4$ is repeated }} {{eqn | ll = 258:       | l = 258 \times 2       | r = 516       | c = $5$ is repeated }} {{eqn | ll = 261:       | l = 261 \times 2       | r = 522       | c = $2$ is repeated }} {{eqn | ll = 264:       | l = 264 \times 2       | r = 528       | c = $2$ is repeated }} {{eqn | ll = 267:       | l = 267 \times 3       | r = 801       | c = $0$ is used }} {{eqn | ll = 273:       | l = 273 + 546       | r = 819       | c = is the $3$rd solution }} {{eqn | ll = 276:       | l = 276 \times 2       | r = 552       | c = $2, 5$ are repeated }} {{eqn | ll = 279:       | l = 279 \times 3       | r = 837       | c = $7$ is repeated }} {{eqn | ll = 291:       | l = 291 \times 2       | r = 582       | c = $2$ is repeated }} {{eqn | ll = 294:       | l = 294 \times 2       | r = 588       | c = $8$ is repeated }} {{eqn | ll = 297:       | l = 297 \times 2       | r = 594       | c = $9$ is repeated }} {{eqn | ll = 312:       | l = 312 \times 2       | r = 624       | c = $2$ is repeated }} {{eqn | ll = 318:       | l = 318 \times 2       | r = 636       | c = $3, 6$ are repeated }} {{eqn | ll = 321:       | l = 321 \times 2       | r = 642       | c = $2$ is repeated }} {{eqn | ll = 324:       | l = 324 \times 2       | r = 648       | c = $4$ is repeated }} {{eqn | ll = 327:       | l = 327 + 654       | r = 981       | c = is the $4$th solution }} {{end-eqn}} Hence the result. {{qed}}