Henry Ernest Dudeney/Puzzles and Curious Problems/108 - The Nine Digits/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $108$

The Nine Digits
It will be found that $32 \, 547 \, 891$ multiplied by $6$ (thus using all the $9$ digits once, and once only)
gives the product $195 \, 287 \, 346$ (also containing all the $9$ digits once, and once only).
Can you find another number to be multiplied by $6$ under the same conditions?


Solution

Dudeney offers up:

$94 \, 857 \, 312 \times 6 = 569 \, 143 \, 872$

Martin Gardner reports that Victor Meally has found $2$ more:

$89 \, 745 \, 321 \times 6 = 538 \, 471 \, 926$
$98 \, 745 \, 231 \times 6 = 592 \, 471 \, 386$


However, an exhaustive search using a simple computer program reveals that there are in fact $87$ solutions in total:

\(\ds 21578943 \times 6\) \(=\) \(\ds 129473658\)
\(\ds 23158794 \times 6\) \(=\) \(\ds 138952764\)
\(\ds 24598731 \times 6\) \(=\) \(\ds 147592386\)
\(\ds 24958731 \times 6\) \(=\) \(\ds 149752386\)
\(\ds 27548913 \times 6\) \(=\) \(\ds 165293478\)
\(\ds 27891543 \times 6\) \(=\) \(\ds 167349258\)
\(\ds 27893154 \times 6\) \(=\) \(\ds 167358924\)
\(\ds 28731594 \times 6\) \(=\) \(\ds 172389564\)
\(\ds 28943157 \times 6\) \(=\) \(\ds 173658942\)
\(\ds 29415873 \times 6\) \(=\) \(\ds 176495238\)
\(\ds 31275489 \times 6\) \(=\) \(\ds 187652934\)
\(\ds 31542789 \times 6\) \(=\) \(\ds 189256734\)
\(\ds 31578942 \times 6\) \(=\) \(\ds 189473652\)
\(\ds 31587294 \times 6\) \(=\) \(\ds 189523764\)
\(\ds 32458971 \times 6\) \(=\) \(\ds 194753826\)
\(\ds 32547891 \times 6\) \(=\) \(\ds 195287346\)
\(\ds 32714589 \times 6\) \(=\) \(\ds 196287534\)
\(\ds 32897541 \times 6\) \(=\) \(\ds 197385246\)
\(\ds 41527893 \times 6\) \(=\) \(\ds 249167358\)
\(\ds 41957283 \times 6\) \(=\) \(\ds 251743698\)
\(\ds 41957328 \times 6\) \(=\) \(\ds 251743968\)
\(\ds 41957823 \times 6\) \(=\) \(\ds 251746938\)
\(\ds 41958273 \times 6\) \(=\) \(\ds 251749638\)
\(\ds 42195783 \times 6\) \(=\) \(\ds 253174698\)
\(\ds 42319578 \times 6\) \(=\) \(\ds 253917468\)
\(\ds 42719583 \times 6\) \(=\) \(\ds 256317498\)
\(\ds 42731958 \times 6\) \(=\) \(\ds 256391748\)
\(\ds 42789153 \times 6\) \(=\) \(\ds 256734918\)
\(\ds 42819573 \times 6\) \(=\) \(\ds 256917438\)
\(\ds 42985731 \times 6\) \(=\) \(\ds 257914386\)
\(\ds 43152789 \times 6\) \(=\) \(\ds 258916734\)
\(\ds 43195728 \times 6\) \(=\) \(\ds 259174368\)
\(\ds 43219578 \times 6\) \(=\) \(\ds 259317468\)
\(\ds 43271958 \times 6\) \(=\) \(\ds 259631748\)
\(\ds 45719283 \times 6\) \(=\) \(\ds 274315698\)
\(\ds 45719328 \times 6\) \(=\) \(\ds 274315968\)
\(\ds 45728193 \times 6\) \(=\) \(\ds 274369158\)
\(\ds 45731928 \times 6\) \(=\) \(\ds 274391568\)
\(\ds 45781923 \times 6\) \(=\) \(\ds 274691538\)
\(\ds 45782193 \times 6\) \(=\) \(\ds 274693158\)
\(\ds 45819273 \times 6\) \(=\) \(\ds 274915638\)
\(\ds 45827193 \times 6\) \(=\) \(\ds 274963158\)
\(\ds 47328591 \times 6\) \(=\) \(\ds 283971546\)
\(\ds 47532891 \times 6\) \(=\) \(\ds 285197346\)
\(\ds 48572931 \times 6\) \(=\) \(\ds 291437586\)
\(\ds 48579231 \times 6\) \(=\) \(\ds 291475386\)
\(\ds 48591273 \times 6\) \(=\) \(\ds 291547638\)
\(\ds 48912753 \times 6\) \(=\) \(\ds 293476518\)
\(\ds 49285731 \times 6\) \(=\) \(\ds 295714386\)
\(\ds 52487931 \times 6\) \(=\) \(\ds 314927586\)
\(\ds 52874931 \times 6\) \(=\) \(\ds 317249586\)
\(\ds 52987431 \times 6\) \(=\) \(\ds 317924586\)
\(\ds 71528943 \times 6\) \(=\) \(\ds 429173658\)
\(\ds 71954283 \times 6\) \(=\) \(\ds 431725698\)
\(\ds 71954328 \times 6\) \(=\) \(\ds 431725968\)
\(\ds 72819543 \times 6\) \(=\) \(\ds 436917258\)
\(\ds 72854931 \times 6\) \(=\) \(\ds 437129586\)
\(\ds 72985431 \times 6\) \(=\) \(\ds 437912586\)
\(\ds 73195428 \times 6\) \(=\) \(\ds 439172568\)
\(\ds 78195423 \times 6\) \(=\) \(\ds 469172538\)
\(\ds 78219543 \times 6\) \(=\) \(\ds 469317258\)
\(\ds 78549231 \times 6\) \(=\) \(\ds 471295386\)
\(\ds 78942153 \times 6\) \(=\) \(\ds 473652918\)
\(\ds 78943152 \times 6\) \(=\) \(\ds 473658912\)
\(\ds 79854231 \times 6\) \(=\) \(\ds 479125386\)
\(\ds 81954273 \times 6\) \(=\) \(\ds 491725638\)
\(\ds 82719543 \times 6\) \(=\) \(\ds 496317258\)
\(\ds 85473291 \times 6\) \(=\) \(\ds 512839746\)
\(\ds 85491273 \times 6\) \(=\) \(\ds 512947638\)
\(\ds 87249531 \times 6\) \(=\) \(\ds 523497186\)
\(\ds 87294153 \times 6\) \(=\) \(\ds 523764918\)
\(\ds 87315294 \times 6\) \(=\) \(\ds 523891764\)
\(\ds 87495231 \times 6\) \(=\) \(\ds 524971386\)
\(\ds 87941523 \times 6\) \(=\) \(\ds 527649138\)
\(\ds 89532471 \times 6\) \(=\) \(\ds 537194826\)
\(\ds 89532714 \times 6\) \(=\) \(\ds 537196284\)
\(\ds 89745321 \times 6\) \(=\) \(\ds 538471926\)
\(\ds 89145327 \times 6\) \(=\) \(\ds 534871962\)
\(\ds 94152873 \times 6\) \(=\) \(\ds 564917238\)
\(\ds 94857123 \times 6\) \(=\) \(\ds 569142738\)
\(\ds 94857213 \times 6\) \(=\) \(\ds 569143278\)
\(\ds 94857312 \times 6\) \(=\) \(\ds 569143872\)
\(\ds 95248731 \times 6\) \(=\) \(\ds 571492386\)
\(\ds 97328541 \times 6\) \(=\) \(\ds 583971246\)
\(\ds 98541273 \times 6\) \(=\) \(\ds 591247638\)
\(\ds 98724531 \times 6\) \(=\) \(\ds 592347186\)
\(\ds 98745231 \times 6\) \(=\) \(\ds 592471386\)


Try it online!


Sources

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