perplexus.info :: Weights and Scales : 6 coins

6 coins (Posted on 2016-01-13)

There are 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same, except for their labels. The labels are supposed to display the weights of the coins.

How can you determine whether all the labels are correct,
using the balance scale only twice?

See The Solution Submitted by Ady TZIDON

Rating: 4.2500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

computer solution

| Comment 5 of 10 |

1 13 5 125 36

123456

Weigh those labeled 1 and 3 against that labeled 5. The RHS should be heavy if it's accurately labeled.

Then weigh those labeled 1, 2 and 5 against those labeled 3 and 6. Again the RHS should go down.

If it passes these tests the labelling is good.

Program modified to allow unequal weights, placing any given weight on either the right, left or neither pan, twice. The program is inefficient, in finding basically the same solution eight times, with pans reversed or weighings reversed, so run time was slowed (but development time was saved).

DefDbl A-Z

Dim crlf$, combs(21, 10) As String, ncombs(21), how$(720)

Private Sub Form_Load()

Form1.Visible = True

Text1.Text = ""

crlf = Chr$(13) + Chr$(10)

For n1 = 0 To 1

For n2 = 0 To 1

For n3 = 0 To 1

For n4 = 0 To 1

For n5 = 0 To 1

For n6 = 0 To 1

tot = n1 + 2 * n2 + 3 * n3 + 4 * n4 + 5 * n5 + 6 * n6

cm$ = ""

If n1 Then cm = cm + "1"

If n2 Then cm = cm + "2"

If n3 Then cm = cm + "3"

If n4 Then cm = cm + "4"

If n5 Then cm = cm + "5"

If n6 Then cm = cm + "6"

ncombs(tot) = ncombs(tot) + 1

combs(tot, ncombs(tot)) = cm

For tot = 1 To 21

For i = 1 To ncombs(tot)

Text1.Text = Text1.Text & mform(tot, "##0") & " " & combs(tot, i) & crlf

Text1.Text = Text1.Text & crlf

minways = 9999

For n1a = 0 To 2

Text1.Text = Text1.Text & n1a & crlf

For n2a = 0 To 2

For n3a = 0 To 2

For n4a = 0 To 2

For n5a = 0 To 2

For n6a = 0 To 2

For n1b = 0 To 2

For n2b = 0 To 2

For n3b = 0 To 2

For n4b = 0 To 2

For n5b = 0 To 2

For n6b = 0 To 2

lhs1$ = "": rhs1$ = "": lhs2$ = "": rhs2$ = ""

exp1l = 0: exp2l = 0: exp1r = 0: exp2r = 0

Select Case n1a

Case 1: lhs1 = lhs1 + "1": exp1l = exp1l + 1

Case 2: rhs1 = rhs1 + "1": exp1r = exp1r + 1

End Select

Select Case n2a

Case 1: lhs1 = lhs1 + "2": exp1l = exp1l + 2

Case 2: rhs1 = rhs1 + "2": exp1r = exp1r + 2

End Select

Select Case n3a

Case 1: lhs1 = lhs1 + "3": exp1l = exp1l + 3

Case 2: rhs1 = rhs1 + "3": exp1r = exp1r + 3

End Select

Select Case n4a

Case 1: lhs1 = lhs1 + "4": exp1l = exp1l + 4

Case 2: rhs1 = rhs1 + "4": exp1r = exp1r + 4

End Select

Select Case n5a

Case 1: lhs1 = lhs1 + "5": exp1l = exp1l + 5

Case 2: rhs1 = rhs1 + "5": exp1r = exp1r + 5

End Select

Select Case n6a

Case 1: lhs1 = lhs1 + "6": exp1l = exp1l + 6

Case 2: rhs1 = rhs1 + "6": exp1r = exp1r + 6

End Select

Select Case n1b

Case 1: lhs2 = lhs2 + "1": exp2l = exp2l + 1

Case 2: rhs2 = rhs2 + "1": exp2r = exp2r + 1

End Select

Select Case n2b

Case 1: lhs2 = lhs2 + "2": exp2l = exp2l + 2

Case 2: rhs2 = rhs2 + "2": exp2r = exp2r + 2

End Select

Select Case n3b

Case 1: lhs2 = lhs2 + "3": exp2l = exp2l + 3

Case 2: rhs2 = rhs2 + "3": exp2r = exp2r + 3

End Select

Select Case n4b

Case 1: lhs2 = lhs2 + "4": exp2l = exp2l + 4

Case 2: rhs2 = rhs2 + "4": exp2r = exp2r + 4

End Select

Select Case n5b

Case 1: lhs2 = lhs2 + "5": exp2l = exp2l + 5

Case 2: rhs2 = rhs2 + "5": exp2r = exp2r + 5

End Select

Select Case n6b

Case 1: lhs2 = lhs2 + "6": exp2l = exp2l + 6

Case 2: rhs2 = rhs2 + "6": exp2r = exp2r + 6

End Select

rhsh1 = Sgn(exp1r - exp1l)

rhsh2 = Sgn(exp2r - exp2l)

ws = ways(lhs1, rhs1, lhs2, rhs2, rhsh1, rhsh2)

If ws < minways And ws > 0 Or ws = 1 Then

minways = ws

Text1.Text = Text1.Text & minways & " " & lhs1 & " " & rhs1 & " " & lhs2 & " " & rhs2 & crlf

If minways < 10 Then

For j = 1 To minways

Text1.Text = Text1.Text & how(j) & crlf

Text1.Text = Text1.Text & crlf

End If

If ws = 1 Then

Text1.Text = Text1.Text & lhs1 & " " & rhs1 & " " & lhs2 & " " & rhs2 & crlf

End If

Text1.Text = Text1.Text & crlf & minways & " done"

Exit Sub

coins$ = "123456": h$ = coins

DoEvents

lhst1 = Val(Mid(coins, 1, 1)) + Val(Mid(coins, 2, 1)) + Val(Mid(coins, 4, 1))

rhst1 = Val(Mid(coins, 5, 1)) + Val(Mid(coins, 2, 1))

lhst2 = Val(Mid(coins, 1, 1)) + Val(Mid(coins, 6, 1))

rhst2 = Val(Mid(coins, 3, 1)) + Val(Mid(coins, 4, 1))

If lhst1 = rhst1 And lhst2 = rhst2 Then

Text1.Text = Text1.Text & coins & crlf

End If

permute coins

Loop Until coins = h

Text1.Text = Text1.Text & crlf & " done"

End Sub

Function mform$(x, t$)

a$ = Format$(x, t$)

If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$

mform$ = a$

End Function

Function ways(a$, b$, c$, d$, rhi1, rhi2)

w = 0

coins$ = "123456": h$ = coins

DoEvents

lhst1 = 0

For i = 1 To Len(a)

lhst1 = lhst1 + Val(Mid(coins, Val(Mid(a, i, 1)), 1))

rhst1 = 0

For i = 1 To Len(b)

rhst1 = rhst1 + Val(Mid(coins, Val(Mid(b, i, 1)), 1))

lhst2 = 0

For i = 1 To Len(c)

lhst2 = lhst2 + Val(Mid(coins, Val(Mid(c, i, 1)), 1))

rhst2 = 0

For i = 1 To Len(d)

rhst2 = rhst2 + Val(Mid(coins, Val(Mid(d, i, 1)), 1))

If Sgn(rhst1 - lhst1) = rhi1 And Sgn(rhst2 - lhst2) = rhi2 Then

w = w + 1

how$(w) = coins$

End If

permute coins

Loop Until coins = h

ways = w

End Function

verification via

lhs1$ = "13": rhs1$ = "5"

lhs2$ = "125": rhs2$ = "36"

rhsh1 = 1: rhsh2 = 1

ws = ways(lhs1, rhs1, lhs2, rhs2, rhsh1, rhsh2)

resulting pan dips show only 123456 results in R R.

123456 R R

123465 R L

123546 = R

123564 R L

123645 = R

123654 R L

124356 = R

124365 R =

124536 L R

124563 R L

124635 L R

124653 = L

125346 L R

125364 = =

125436 L R

125463 = L

125634 L R

125643 L R

126345 L R

126354 L R

126435 L R

126453 L R

126534 L R

126543 L R

132456 R L

132465 R L

132546 R =

132564 R L

132645 R L

132654 R L

134256 = R

134265 R L

134526 L R

134562 R L

134625 L R

134652 = L

135246 L R

135264 = L

135426 L R

135462 = L

135624 L R

135642 L L

136245 L R

136254 L R

136425 L R

136452 L L

136524 L R

136542 L =

142356 R L

142365 R L

142536 = =

142563 R L

142635 = L

142653 R L

143256 R L

143265 R L

143526 L R

143562 R L

143625 L R

143652 R L

145236 L R

145263 = L

145326 L R

145362 = L

145623 L R

145632 L L

146235 L R

146253 L L

146325 L R

146352 L L

146523 L R

146532 L =

152346 R L

152364 R L

152436 = L

152463 R L

152634 = L

152643 R L

153246 = L

153264 R L

153426 L R

153462 R L

153624 L L

153642 = L

154236 L R

154263 R L

154326 L R

154362 R L

154623 L L

154632 L L

156234 L R

156243 L L

156324 L R

156342 L L

156423 L R

156432 L L

162345 R L

162354 R L

162435 = L

162453 R L

162534 = L

162543 R L

163245 = L

163254 R L

163425 L L

163452 R L

163524 L L

163542 = L

164235 L L

164253 = L

164325 L =

164352 = L

164523 L L

164532 L L

165234 L L

165243 L L

165324 L =

165342 L L

165423 L L

165432 L L

213456 = R

213465 R L

213546 L R

213564 R L

213645 L R

213654 = L

214356 L R

214365 = =

214536 L R

214563 = L

214635 L R

214653 L L

215346 L R

215364 L =

215436 L R

215463 L L

215634 L R

215643 L R

216345 L R

216354 L R

216435 L R

216453 L R

216534 L R

216543 L R

231456 R L

231465 R L

231546 R L

231564 R L

231645 R L

231654 R L

234156 L =

234165 = L

234516 L R

234561 = L

234615 L R

234651 L L

235146 L R

235164 L L

235416 L R

235461 L L

235614 L R

235641 L L

236145 L R

236154 L =

236415 L R

236451 L L

236514 L R

236541 L L

241356 R L

241365 R L

241536 = L

241563 R L

241635 = L

241653 R L

243156 = L

243165 R L

243516 L R

243561 R L

243615 L R

243651 = L

245136 L R

245163 L L

245316 L R

245361 L L

245613 L R

245631 L L

246135 L R

246153 L L

246315 L R

246351 L L

246513 L R

246531 L L

251346 R L

251364 R L

251436 = L

251463 R L

251634 = L

251643 R L

253146 L L

253164 R L

253416 L R

253461 R L

253614 L L

253641 L L

254136 L =

254163 = L

254316 L R

254361 = L

254613 L L

254631 L L

256134 L =

256143 L L

256314 L R

256341 L L

256413 L R

256431 L L

261345 R L

261354 R L

261435 = L

261453 R L

261534 = L

261543 R L

263145 L L

263154 = L

263415 L L

263451 = L

263514 L L

263541 L L

264135 L L

264153 L L

264315 L =

264351 L L

264513 L L

264531 L L

265134 L L

265143 L L

265314 L =

265341 L L

265413 L L

265431 L L

312456 = L

312465 R L

312546 L =

312564 R L

312645 L L

312654 = L

314256 L R

314265 L L

314526 L R

314562 L L

314625 L R

314652 L L

315246 L R

315264 L L

315426 L R

315462 L L

315624 L R

315642 L L

316245 L R

316254 L R

316425 L R

316452 L L

316524 L R

316542 L =

321456 R L

321465 R L

321546 = L

321564 R L

321645 = L

321654 R L

324156 L =

324165 L L

324516 L R

324561 L L

324615 L R

324651 L L

325146 L R

325164 L L

325416 L R

325461 L L

325614 L R

325641 L L

326145 L R

326154 L =

326415 L R

326451 L L

326514 L R

326541 L L

341256 R L

341265 R L

341526 L L

341562 R L

341625 L L

341652 R L

342156 = L

342165 R L

342516 L =

342561 R L

342615 L L

342651 = L

345126 L R

345162 L L

345216 L R

345261 L L

345612 L L

345621 L L

346125 L R

346152 L L

346215 L R

346251 L L

346512 L =

346521 L L

351246 = L

351264 R L

351426 L L

351462 R L

351624 L L

351642 = L

352146 L L

352164 R L

352416 L L

352461 R L

352614 L L

352641 L L

354126 L =

354162 L L

354216 L R

354261 L L

354612 L L

354621 L L

356124 L =

356142 L L

356214 L R

356241 L L

356412 L L

356421 L L

361245 = L

361254 R L

361425 L L

361452 R L

361524 L L

361542 = L

362145 L L

362154 = L

362415 L L

362451 = L

362514 L L

362541 L L

364125 L L

364152 L L

364215 L L

364251 L L

364512 L L

364521 L L

365124 L L

365142 L L

365214 L L

365241 L L

365412 L L

365421 L L

412356 L L

412365 = L

412536 L =

412563 = L

412635 L L

412653 L L

413256 L L

413265 L L

413526 L R

413562 L L

413625 L R

413652 L L

415236 L R

415263 L L

415326 L R

415362 L L

415623 L R

415632 L L

416235 L R

416253 L L

416325 L R

416352 L L

416523 L R

416532 L =

421356 = L

421365 R L

421536 L L

421563 R L

421635 L L

421653 = L

423156 L L

423165 L L

423516 L R

423561 L L

423615 L R

423651 L L

425136 L R

425163 L L

425316 L R

425361 L L

425613 L R

425631 L L

426135 L R

426153 L L

426315 L R

426351 L L

426513 L R

426531 L L

431256 = L

431265 R L

431526 L L

431562 R L

431625 L L

431652 = L

432156 L L

432165 = L

432516 L =

432561 = L

432615 L L

432651 L L

435126 L R

435162 L L

435216 L R

435261 L L

435612 L L

435621 L L

436125 L R

436152 L L

436215 L R

436251 L L

436512 L =

436521 L L

451236 L L

451263 R L

451326 L L

451362 R L

451623 L L

451632 L L

452136 L L

452163 = L

452316 L L

452361 = L

452613 L L

452631 L L

453126 L L

453162 L L

453216 L L

453261 L L

453612 L L

453621 L L

456123 L L

456132 L L

456213 L L

456231 L L

456312 L L

456321 L L

461235 L L

461253 = L

461325 L L

461352 = L

461523 L L

461532 L L

462135 L L

462153 L L

462315 L L

462351 L L

462513 L L

462531 L L

463125 L L

463152 L L

463215 L L

463251 L L

463512 L L

463521 L L

465123 L L

465132 L L

465213 L L

465231 L L

465312 L L

465321 L L

512346 L L

512364 L L

512436 L L

512463 L L

512634 L L

512643 L L

513246 L L

513264 L L

513426 L R

513462 L L

513624 L L

513642 L L

514236 L R

514263 L L

514326 L R

514362 L L

514623 L L

514632 L L

516234 L R

516243 L L

516324 L R

516342 L L

516423 L R

516432 L L

521346 L L

521364 = L

521436 L L

521463 = L

521634 L L

521643 L L

523146 L L

523164 L L

523416 L R

523461 L L

523614 L L

523641 L L

524136 L =

524163 L L

524316 L R

524361 L L

524613 L L

524631 L L

526134 L =

526143 L L

526314 L R

526341 L L

526413 L R

526431 L L

531246 L L

531264 = L

531426 L L

531462 = L

531624 L L

531642 L L

532146 L L

532164 L L

532416 L L

532461 L L

532614 L L

532641 L L

534126 L =

534162 L L

534216 L R

534261 L L

534612 L L

534621 L L

536124 L =

536142 L L

536214 L R

536241 L L

536412 L L

536421 L L

541236 L L

541263 = L

541326 L L

541362 = L

541623 L L

541632 L L

542136 L L

542163 L L

542316 L L

542361 L L

542613 L L

542631 L L

543126 L L

543162 L L

543216 L L

543261 L L

543612 L L

543621 L L

546123 L L

546132 L L

546213 L L

546231 L L

546312 L L

546321 L L

561234 L L

561243 L L

561324 L L

561342 L L

561423 L L

561432 L L

562134 L L

562143 L L

562314 L L

562341 L L

562413 L L

562431 L L

563124 L L

563142 L L

563214 L L

563241 L L

563412 L L

563421 L L

564123 L L

564132 L L

564213 L L

564231 L L

564312 L L

564321 L L

612345 L L

612354 L L

612435 L L

612453 L L

612534 L L

612543 L L

613245 L L

613254 L L

613425 L L

613452 L L

613524 L L

613542 L L

614235 L L

614253 L L

614325 L =

614352 L L

614523 L L

614532 L L

615234 L L

615243 L L

615324 L =

615342 L L

615423 L L

615432 L L

621345 L L

621354 L L

621435 L L

621453 L L

621534 L L

621543 L L

623145 L L

623154 L L

623415 L L

623451 L L

623514 L L

623541 L L

624135 L L

624153 L L

624315 L =

624351 L L

624513 L L

624531 L L

625134 L L

625143 L L

625314 L =

625341 L L

625413 L L

625431 L L

631245 L L

631254 L L

631425 L L

631452 L L

631524 L L

631542 L L

632145 L L

632154 L L

632415 L L

632451 L L

632514 L L

632541 L L

634125 L L

634152 L L

634215 L L

634251 L L

634512 L L

634521 L L

635124 L L

635142 L L

635214 L L

635241 L L

635412 L L

635421 L L

641235 L L

641253 L L

641325 L L

641352 L L

641523 L L

641532 L L

642135 L L

642153 L L

642315 L L

642351 L L

642513 L L

642531 L L

643125 L L

643152 L L

643215 L L

643251 L L

643512 L L

643521 L L

645123 L L

645132 L L

645213 L L

645231 L L

645312 L L

645321 L L

651234 L L

651243 L L

651324 L L

651342 L L

651423 L L

651432 L L

652134 L L

652143 L L

652314 L L

652341 L L

652413 L L

652431 L L

653124 L L

653142 L L

653214 L L

653241 L L

653412 L L

653421 L L

654123 L L

654132 L L

654213 L L

654231 L L

654312 L L

654321 L L

Posted by Charlie on 2016-01-13 14:32:07

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