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:
\(\ds 21578943 \times 6\) | \(=\) | \(\ds 129473658\) | ||||||||||||
\(\ds 23158794 \times 6\) | \(=\) | \(\ds 138952764\) | ||||||||||||
\(\ds 24598731 \times 6\) | \(=\) | \(\ds 147592386\) | ||||||||||||
\(\ds 24958731 \times 6\) | \(=\) | \(\ds 149752386\) | ||||||||||||
\(\ds 27548913 \times 6\) | \(=\) | \(\ds 165293478\) | ||||||||||||
\(\ds 27891543 \times 6\) | \(=\) | \(\ds 167349258\) | ||||||||||||
\(\ds 27893154 \times 6\) | \(=\) | \(\ds 167358924\) | ||||||||||||
\(\ds 28731594 \times 6\) | \(=\) | \(\ds 172389564\) | ||||||||||||
\(\ds 28943157 \times 6\) | \(=\) | \(\ds 173658942\) | ||||||||||||
\(\ds 29415873 \times 6\) | \(=\) | \(\ds 176495238\) | ||||||||||||
\(\ds 31275489 \times 6\) | \(=\) | \(\ds 187652934\) | ||||||||||||
\(\ds 31542789 \times 6\) | \(=\) | \(\ds 189256734\) | ||||||||||||
\(\ds 31578942 \times 6\) | \(=\) | \(\ds 189473652\) | ||||||||||||
\(\ds 31587294 \times 6\) | \(=\) | \(\ds 189523764\) | ||||||||||||
\(\ds 32458971 \times 6\) | \(=\) | \(\ds 194753826\) | ||||||||||||
\(\ds 32547891 \times 6\) | \(=\) | \(\ds 195287346\) | ||||||||||||
\(\ds 32714589 \times 6\) | \(=\) | \(\ds 196287534\) | ||||||||||||
\(\ds 32897541 \times 6\) | \(=\) | \(\ds 197385246\) | ||||||||||||
\(\ds 41527893 \times 6\) | \(=\) | \(\ds 249167358\) | ||||||||||||
\(\ds 41957283 \times 6\) | \(=\) | \(\ds 251743698\) | ||||||||||||
\(\ds 41957328 \times 6\) | \(=\) | \(\ds 251743968\) | ||||||||||||
\(\ds 41957823 \times 6\) | \(=\) | \(\ds 251746938\) | ||||||||||||
\(\ds 41958273 \times 6\) | \(=\) | \(\ds 251749638\) | ||||||||||||
\(\ds 42195783 \times 6\) | \(=\) | \(\ds 253174698\) | ||||||||||||
\(\ds 42319578 \times 6\) | \(=\) | \(\ds 253917468\) | ||||||||||||
\(\ds 42719583 \times 6\) | \(=\) | \(\ds 256317498\) | ||||||||||||
\(\ds 42731958 \times 6\) | \(=\) | \(\ds 256391748\) | ||||||||||||
\(\ds 42789153 \times 6\) | \(=\) | \(\ds 256734918\) | ||||||||||||
\(\ds 42819573 \times 6\) | \(=\) | \(\ds 256917438\) | ||||||||||||
\(\ds 42985731 \times 6\) | \(=\) | \(\ds 257914386\) | ||||||||||||
\(\ds 43152789 \times 6\) | \(=\) | \(\ds 258916734\) | ||||||||||||
\(\ds 43195728 \times 6\) | \(=\) | \(\ds 259174368\) | ||||||||||||
\(\ds 43219578 \times 6\) | \(=\) | \(\ds 259317468\) | ||||||||||||
\(\ds 43271958 \times 6\) | \(=\) | \(\ds 259631748\) | ||||||||||||
\(\ds 45719283 \times 6\) | \(=\) | \(\ds 274315698\) | ||||||||||||
\(\ds 45719328 \times 6\) | \(=\) | \(\ds 274315968\) | ||||||||||||
\(\ds 45728193 \times 6\) | \(=\) | \(\ds 274369158\) | ||||||||||||
\(\ds 45731928 \times 6\) | \(=\) | \(\ds 274391568\) | ||||||||||||
\(\ds 45781923 \times 6\) | \(=\) | \(\ds 274691538\) | ||||||||||||
\(\ds 45782193 \times 6\) | \(=\) | \(\ds 274693158\) | ||||||||||||
\(\ds 45819273 \times 6\) | \(=\) | \(\ds 274915638\) | ||||||||||||
\(\ds 45827193 \times 6\) | \(=\) | \(\ds 274963158\) | ||||||||||||
\(\ds 47328591 \times 6\) | \(=\) | \(\ds 283971546\) | ||||||||||||
\(\ds 47532891 \times 6\) | \(=\) | \(\ds 285197346\) | ||||||||||||
\(\ds 48572931 \times 6\) | \(=\) | \(\ds 291437586\) | ||||||||||||
\(\ds 48579231 \times 6\) | \(=\) | \(\ds 291475386\) | ||||||||||||
\(\ds 48591273 \times 6\) | \(=\) | \(\ds 291547638\) | ||||||||||||
\(\ds 48912753 \times 6\) | \(=\) | \(\ds 293476518\) | ||||||||||||
\(\ds 49285731 \times 6\) | \(=\) | \(\ds 295714386\) | ||||||||||||
\(\ds 52487931 \times 6\) | \(=\) | \(\ds 314927586\) | ||||||||||||
\(\ds 52874931 \times 6\) | \(=\) | \(\ds 317249586\) | ||||||||||||
\(\ds 52987431 \times 6\) | \(=\) | \(\ds 317924586\) | ||||||||||||
\(\ds 71528943 \times 6\) | \(=\) | \(\ds 429173658\) | ||||||||||||
\(\ds 71954283 \times 6\) | \(=\) | \(\ds 431725698\) | ||||||||||||
\(\ds 71954328 \times 6\) | \(=\) | \(\ds 431725968\) | ||||||||||||
\(\ds 72819543 \times 6\) | \(=\) | \(\ds 436917258\) | ||||||||||||
\(\ds 72854931 \times 6\) | \(=\) | \(\ds 437129586\) | ||||||||||||
\(\ds 72985431 \times 6\) | \(=\) | \(\ds 437912586\) | ||||||||||||
\(\ds 73195428 \times 6\) | \(=\) | \(\ds 439172568\) | ||||||||||||
\(\ds 78195423 \times 6\) | \(=\) | \(\ds 469172538\) | ||||||||||||
\(\ds 78219543 \times 6\) | \(=\) | \(\ds 469317258\) | ||||||||||||
\(\ds 78549231 \times 6\) | \(=\) | \(\ds 471295386\) | ||||||||||||
\(\ds 78942153 \times 6\) | \(=\) | \(\ds 473652918\) | ||||||||||||
\(\ds 78943152 \times 6\) | \(=\) | \(\ds 473658912\) | ||||||||||||
\(\ds 79854231 \times 6\) | \(=\) | \(\ds 479125386\) | ||||||||||||
\(\ds 81954273 \times 6\) | \(=\) | \(\ds 491725638\) | ||||||||||||
\(\ds 82719543 \times 6\) | \(=\) | \(\ds 496317258\) | ||||||||||||
\(\ds 85473291 \times 6\) | \(=\) | \(\ds 512839746\) | ||||||||||||
\(\ds 85491273 \times 6\) | \(=\) | \(\ds 512947638\) | ||||||||||||
\(\ds 87249531 \times 6\) | \(=\) | \(\ds 523497186\) | ||||||||||||
\(\ds 87294153 \times 6\) | \(=\) | \(\ds 523764918\) | ||||||||||||
\(\ds 87315294 \times 6\) | \(=\) | \(\ds 523891764\) | ||||||||||||
\(\ds 87495231 \times 6\) | \(=\) | \(\ds 524971386\) | ||||||||||||
\(\ds 87941523 \times 6\) | \(=\) | \(\ds 527649138\) | ||||||||||||
\(\ds 89532471 \times 6\) | \(=\) | \(\ds 537194826\) | ||||||||||||
\(\ds 89532714 \times 6\) | \(=\) | \(\ds 537196284\) | ||||||||||||
\(\ds 89745321 \times 6\) | \(=\) | \(\ds 538471926\) | ||||||||||||
\(\ds 89145327 \times 6\) | \(=\) | \(\ds 534871962\) | ||||||||||||
\(\ds 94152873 \times 6\) | \(=\) | \(\ds 564917238\) | ||||||||||||
\(\ds 94857123 \times 6\) | \(=\) | \(\ds 569142738\) | ||||||||||||
\(\ds 94857213 \times 6\) | \(=\) | \(\ds 569143278\) | ||||||||||||
\(\ds 94857312 \times 6\) | \(=\) | \(\ds 569143872\) | ||||||||||||
\(\ds 95248731 \times 6\) | \(=\) | \(\ds 571492386\) | ||||||||||||
\(\ds 97328541 \times 6\) | \(=\) | \(\ds 583971246\) | ||||||||||||
\(\ds 98541273 \times 6\) | \(=\) | \(\ds 591247638\) | ||||||||||||
\(\ds 98724531 \times 6\) | \(=\) | \(\ds 592347186\) | ||||||||||||
\(\ds 98745231 \times 6\) | \(=\) | \(\ds 592471386\) |
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $108$. -- The Nine Digits
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $136$. The Nine Digits