519:
32:
758:\// A light /\\ A heavy /\/ B light \/\ B heavy //\ C light \\/ C heavy \/– D light /\– D heavy –\/ E light –/\ E heavy /–\ F light \–/ F heavy \\– G light //– G heavy –\\ H light –// H heavy \–\ I light /–/ I heavy /–– J light \–– J heavy –/– K light –\– K heavy ––/ L light ––\ L heavy ––– M either lighter or heavier (13-coin case), or all coins weigh the same (12-coin case)
578:-th digit identical to the label of the plate (with three plates, one on each side of the scale labelled 0 and 2, and one off the scale labelled 1). Other step-by-step procedures are similar to the following. It is less straightforward for this problem, and the second and third weighings depend on what has happened previously, although that need not be the case (see below).
747:
The three possible outcomes of each weighing can be denoted by "\" for the left side being lighter, "/" for the right side being lighter, and "–" for both sides having the same weight. The symbols for the weighings are listed in sequence. For example, "//–" means that the right side is lighter in the
693:
b. If the 3 coins do not balance, then the odd coin is from this population of 3 coins. Pay attention to the direction of the balance swing (up means the odd coin is light, down means it is heavy). Remove one of the 3 coins, move another to the other side of the balance (remove all other coins from
605:
1.b) The side that was heavier the first time is lighter the second time. This means that one of the three coins that went from the lighter side to the heavier side is the light coin. For the third attempt, weigh two of these coins against each other: if one is lighter, it is the unique coin; if they
732:
In a relaxed variation of this puzzle, one only needs to find the counterfeit coin without necessarily being able to tell its weight relative to the others. In this case, clearly any solution that previously weighed every coin at some point can be adapted to handle one extra coin. This coin is never
560:
A more complex version has twelve coins, eleven of twelve of which are identical. If one is different, we don't know whether it is heavier or lighter than the others. This time the balance may be used three times to determine if there is a unique coin—and if there is, to isolate it and determine its
719:
If the scales are only off balance once, then it must be one of the coins 1, 2, 3—which only appear in one weighing. If there is never balance then it must be one of the coins 10–13 that appear in all weighings. Picking out the one counterfeit coin corresponding to each of the 27 outcomes is always
641:
2.b) The three remaining coins are heavier. In this case you now know that one of those three coins is the odd one out and that it is heavier. Take two of those three coins and weigh them against each other. If the balance tips then the heavier coin is the odd one out. If the two coins balance then
631:
2.a) The three remaining coins are lighter. In this case you now know that one of those three coins is the odd one out and that it is lighter. Take two of those three coins and weigh them against each other. If the balance tips then the lighter coin is the odd one out. If the two coins balance then
544:
Now, imagine the nine coins in three stacks of three coins each. In one move we can find which of the three stacks is lighter (i.e. the one containing the lighter coin). It then takes only one more move to identify the light coin from within that lighter stack. So in two weighings, we can find a
540:
To find a solution, we first consider the maximum number of items from which one can find the lighter one in just one weighing. The maximum number possible is three. To find the lighter one, we can compare any two coins, leaving the third out. If the two coins weigh the same, then the lighter coin
595:
1.a) The same side that was heavier the first time is still heavier. This means that either the coin that stayed there is heavier or that the coin that stayed on the lighter side is lighter. Balancing one of these against one of the other ten coins reveals which of these is true, thus solving the
4726:
indistinguishable coins, one of which is fake (it is not known whether it is heavier or lighter than the genuine coins, which all weigh the same). There are two balance scales that can be used in parallel. Each weighing lasts three minute. What is the largest number of coins N for which it is
761:
Using the pattern of outcomes above, the composition of coins for each weighing can be determined; for example the set "\/– D light" implies that coin D must be on the left side in the first weighing (to cause that side to be lighter), on the right side in the second, and unused in the third:
587:
1. One side is heavier than the other. If this is the case, remove three coins from the heavier side, move three coins from the lighter side to the heavier side, and place three coins that were not weighed the first time on the lighter side. (Remember which coins are which.) There are three
754:
As each weighing gives a meaningful result only when the number of coins on the left side is equal to the number on the right side, we disregard the first row, so that each column has the same number of "\" and "/" symbols (four of each). The rows are labelled, the order of the coins being
720:
possible (13 coins one either too heavy or too light is 26 possibilities) except when all weighings are balanced, in which case there is no counterfeit coin (or its weight is correct). If coins 0 and 13 are deleted from these weighings they give one generic solution to the 12-coin problem.
615:
1.c) Both sides are even. This means that one of the three coins that was removed from the heavier side is the heavy coin. For the third attempt, weigh two of these coins against each other: if one is heavier, it is the unique coin; if they balance, the third coin is the heavy
733:
put on the scales, but if all weighings are balanced it is picked as the counterfeit coin. It is not possible to do any better, since any coin that is put on the scales at some point and picked as the counterfeit coin can then always be assigned weight relative to the others.
748:
first and second weighings, and both sides weigh the same in the third weighing. Three weighings give the following 3 = 27 outcomes. Except for "–––", the sets are divided such that each set on the right has a "/" where the set on the left has a "\", and vice versa:
768:
The outcomes are then read off the table. For example, if the right side is lighter in the first two weighings and both sides weigh the same in the third, the corresponding code "//– G heavy" implies that coin G is the odd one, and it is heavier than the others.
689:
a. If the 3 coins balance, then the odd coin is among the remaining population of 2 coins. Test one of the 2 coins against any other coin; if they balance, the odd coin is the last untested coin, if they do not balance, the odd coin is the current test
140:
about balancing items—often coins—to determine which holds a different value, by using balance scales a limited number of times. These differ from puzzles that assign weights to items, in that only the relative mass of these items is relevant.
4344:
Each of these algorithms using 5 weighings finds among 11 coins up to two counterfeit coins which could be heavier or lighter than real coins by the same value. In this case the uncertainty domain (the set of admissible situations) contains
2890:
623:
2. Both sides are even. If this is the case, all eight coins are identical and can be set aside. Take the four remaining coins and place three on one side of the balance. Place 3 of the 8 identical coins on the other side. There are three
694:
balance). If the balance evens out, the odd coin is the coin that was removed. If the balance switches direction, the odd coin is the one that was moved to the other side, otherwise, the odd coin is the coin that remained in place.
1675:
561:
weight relative to the others. (This puzzle and its solution first appeared in an article in 1945.) The problem has a simpler variant with three coins in two weighings, and a more complex variant with 39 coins in four weighings.
3403:
3866:
3245:
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527:
A well-known example has up to nine items, say coins (or balls), that are identical in weight except one, which is lighter than the others—a counterfeit (an oddball). The difference is perceptible only by weighing them on
569:
This problem has more than one solution. One is easily scalable to a higher number of coins by using base-three numbering: labelling each coin with a different number of three digits in base three, and positioning at the
490:= 13. Note that with 3 weighs and 13 coins, it is not always possible to determine the identity of the last coin (whether it is heavier or lighter than the rest), but merely that the coin is different. In general, with
1049:
723:
If two coins are counterfeit, this procedure, in general, does not pick either of these, but rather some authentic coin. For instance, if both coins 1 and 2 are counterfeit, either coin 4 or 5 is wrongly picked.
3568:
3130:
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522:
Solution to the balance puzzle for 9 coins in 2 weighings, where the odd coin is lighter than the others – if the odd coin were heavier than the others, the upper two branches in each weighing decision are
1845:
2177:
1391:
4618:
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3618:
2021:
characterizes the types of objects: the standard type, the non-standard type (i.e., configurations of types), and it does not contain information about relative weights of non-standard objects.
1247:
3306:
2237:
1455:
651:
2.c) The three remaining coins balance. In this case you just need to weigh the remaining coin against any of the other 11 coins and this tells you whether it is heavier, lighter, or the same.
1308:
2618:
2573:
3743:
2525:
705:
If there is one authentic coin for reference then the suspect coins can be thirteen. Number the coins from 1 to 13 and the authentic coin number 0 and perform these weighings in any order:
467:
355:
4193:
2480:
666:
Given a population of 13 coins in which it is known that 1 of the 13 is different (mass) from the rest, it is simple to determine which coin it is with a balance and 3 tests as follows:
249:
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4238:
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2060:
412:
2095:
300:
3660:
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2620:
The incompleteness of initial information about the distribution of weights of a group of objects is characterized by the set of admissible distributions of weights of objects
974:
2441:
2019:
1511:
942:
913:
807:
3470:
3978:
3159:
1749:
4335:
2705:
2378:
1578:
1174:
1141:
2356:
2290:
2323:
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4712:
1990:
1901:
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743:(among 13 coins A–M) find the odd coin, and, at 12/13 probability, tell if it is lighter or heavier (for the remaining 1/13 probability, just that it is different).
203:
4674:
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4516:
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1724:
4283:
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4126:
3998:
3763:
3423:
3179:
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1865:
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881:
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2380:, both pans should contain the same number of objects: if on some pan the number of objects is smaller than as it should be, then it receives some
765:
1st weighing: left side: ADGI, right side: BCFJ 2nd weighing: left side: BEGH, right side: ACDK 3rd weighing: left side: CFHI, right side: ABEL
4341:(Virtakallio-Golay code). At the same time, it is established that a static WA (i.e. weighting code) with the same parameters does not exist.
979:
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552:
By extension, it would take only three weighings to find the odd light coin among 27 coins, and four weighings to find it from 81 coins.
3043:
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1774:
96:
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For example, in detecting a dissimilar coin in three weighings (n = 3), the maximum number of coins that can be analyzed is
68:
2100:
1313:
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4348:
3573:
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740:(among 12 coins A–L) conclude if they all weigh the same, or find the odd coin and tell if it is lighter or heavier, or
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75:
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115:
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418:
306:
4131:
2446:
532:—but only the coins themselves can be weighed. How can one isolate the counterfeit coin with only two weighings?
510:- 1 or less coins. In the case n = 3, you can truly discover the identity of the different coin out of 12 coins.
49:
209:
82:
4624:
for ternary codes) which is, obviously, necessary for the existence of a perfect WA. It is only known that for
4003:
53:
4198:
4066:
686:
3) Test 2a, Test 3 of the coins from the group of 5 coins against any 3 coins from the population of 8 coins:
2027:
751:/// \\\ \// /\\ /\/ \/\ //\ \\/ \/– /\– –\/ –/\ /–\ \–/ \\– //– –\\ –// \–\ /–/ /–– \–– –/– –\– ––/ ––\ –––
369:
2065:
64:
263:
3626:
4813:
Chudnov, Alexander M. (2015). "Weighing algorithms of classification and identification of situations".
3665:
2623:
947:
681:
b. The odd coin is among the population of 8 coins, proceed in the same way as in the 12 coins problem.
541:
must be one of those not on the balance. Otherwise, it is the one indicated as lighter by the balance.
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1995:
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889:
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884:
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1536:
1146:
1113:
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2262:
42:
2885:{\displaystyle W(s|I^{n};\mathrm {h} )=\{\mathrm {x} \in I^{n}|s(\mathrm {x} ;\mathrm {h} )=s\}}
2295:
4894:
4463:
4679:
1959:
1870:
670:
1) Subdivide the coins in to 2 groups of 4 coins and a third group with the remaining 5 coins.
2895:
1926:
4848:
Khovanova, Tanya (2013). "Solution to the
Counterfeit Coin Problem and its Generalization".
4521:
2659:
89:
4460:
To date it is not known whether there are other perfect WA that identify the situations in
2710:
1460:
518:
181:
8:
4653:
4627:
4495:
4434:
1703:
4265:
2741:
1670:{\displaystyle I_{t}^{n}=\{\mathrm {x} \in I^{n}|w(\mathrm {x} )\leq t\}\subseteq I^{n}}
4872:
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678:
a. If the coins balance, the odd coin is in the population of 5 and proceed to test 2a.
4794:
836:
4877:
4834:
3398:{\displaystyle \mathrm {h} ^{j}=\mathrm {A} _{j}(s^{j-1});\mathrm {h} ^{j}\in I^{n},}
736:
A method which weighs the same sets of coins regardless of outcomes lets one either
4822:
3861:{\displaystyle W(s|{\mathcal {A}})=\{\mathrm {z} \in I^{m}|s(z|{\mathcal {A}})=s\}}
20:
830:
3240:{\displaystyle {\mathcal {A}}=<\mathrm {A} _{1},\dots ,\mathrm {A} _{m}>,}
3033:{\displaystyle Z=W(0|Z,\mathrm {h} )+W(1|Z,\mathrm {h} )+W(-1|Z,\mathrm {h} ),}
1698:
529:
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4745:
4621:
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4287:
As an example the perfect dynamic (two-cascade) algorithms with parameters
4262:
an algorithm of identification the types identifies also the situations in
137:
4826:
2656:
which is also called the set of admissible situations, the elements of
2735:
642:
the third coin not on the balance is the odd one out and it is heavier.
632:
the third coin not on the balance is the odd one out and it is lighter.
4337:
are constructed in which correspond to the parameters of the perfect
1044:{\displaystyle \mathrm {e} =(e_{1},\dots ,e_{n})\in \mathbb {R} ^{n}}
364:
Identify if unique coin exists, and whether it is lighter or heavier
3563:{\displaystyle \mathrm {s} ^{j-1}=(s_{1},\dots ,s_{j-1})\in I^{j-1}}
31:
4854:
3125:{\displaystyle W(s|Z,\mathrm {h} )=W(s|I^{n},\mathrm {h} )\cap Z.}
1923:
th object is greater (smaller) by a constant (unknown) value if
1847:
which defines the configurations of weights of the objects: the
1103:{\displaystyle E=\{\mathrm {e} ^{j}\}\subseteq \mathbb {R} ^{n},}
3944:{\displaystyle W(s|Z;{\mathcal {A}})=W(s|{\mathcal {A}})\cap Z.}
361:
Target coin is different from others, or all coins are the same
4714:
which determines the parameters of the constructed perfect WA.
582:
Four coins are put on each side. There are two possibilities:
494:
weighs, you can determine the identity of a coin if you have
777:
The generalization of this problem is described in
Chudnov.
675:
2) Test 1, Test the 2 groups of 4 coins against each other:
1840:{\displaystyle \mathrm {x} =(x_{1},\dots ,x_{n})\in I^{n},}
19:
2239:
has the following interpretation: for a given check the
4879:
Two-pan balance and generalized counterfeit coin problem
574:-th weighing all the coins that are labelled with the
173:
Whether target coin is lighter or heavier than others
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2921:and defines the corresponding partition of the set
2172:{\displaystyle s(\mathrm {x} ;\mathrm {h} )=sign().}
1386:{\displaystyle \mathrm {e} ^{+}=(|sign(e_{i})|)_{i}}
4613:{\displaystyle \sum _{i=0}^{t}2^{i}C_{n}^{i}=3^{m}}
4420:{\displaystyle 1+2C_{11}^{1}+2^{2}C_{11}^{2}=3^{5}}
3613:{\displaystyle \mathrm {h} ^{1}=\mathrm {A} _{1}()}
56:. Unsourced material may be challenged and removed.
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4242:It is proved in that for so-called suitable sets
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1242:{\displaystyle \mathrm {e} ^{*}=(sign(e_{i}))_{i}}
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4676:this equation has the unique nontrivial solution
3980:is said to: a) identify the situations in a set
3745:be the set of situations with the same syndrome
3301:{\displaystyle \mathrm {A} _{j}:I^{j-1}\to I^{n}}
2232:{\displaystyle \mathrm {h} =(h_{1},\dots ,h_{n})}
1450:{\displaystyle E^{+}=\{(\mathrm {e} ^{j})^{+}\}.}
4886:
4427:situations, i.e. the constructed WA lies on the
1303:{\displaystyle E^{*}=\{(\mathrm {e} ^{j})^{*}\}}
4727:possible to find the fake coin in ten minutes?
2613:{\displaystyle s(\mathrm {x} ;\mathrm {h} )=1.}
2568:{\displaystyle s(\mathrm {x} ;\mathrm {h} )=-1}
3738:{\displaystyle W(s|{\mathcal {A}})\subseteq I}
2575:, and the right pan outweighs the left one if
2520:{\displaystyle s(\mathrm {x} ;\mathrm {h} )=0}
2482:describes the following cases: the balance if
4722:Eliran Sabag invented this puzzle. There are
4518:. Moreover, it is not known whether for some
462:{\displaystyle \lceil \log _{3}(2c+3)\rceil }
350:{\displaystyle \lceil \log _{3}(2c+1)\rceil }
4188:{\displaystyle |W^{+}(s|Z{\mathcal {A}})|=1}
3855:
3799:
2879:
2825:
2475:{\displaystyle s(\mathrm {x} ;\mathrm {h} )}
2443:reference objects. The result of a weighing
1651:
1605:
1567:
1546:
1513:; i.e., the set of all sequences of length
1441:
1413:
1297:
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1079:
1064:
456:
422:
344:
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213:
4108:b) identify the types of objects in a set
2259:th object participates in the weighing if
2024:A weighing (a check) is given by a vector
727:
2527:, the left pan outweighs the right one if
244:{\displaystyle \lceil \log _{3}(c)\rceil }
4853:
4847:
4056:{\displaystyle |W(s|Z,{\mathcal {A}})|=1}
2062:the result of a weighing for a situation
1493:
1087:
1031:
953:
789:
606:balance, the third coin is the light one.
116:Learn how and when to remove this message
4233:{\displaystyle s\in S(Z{\mathcal {A}}).}
4101:{\displaystyle s\in S(Z{\mathcal {A}});}
700:
517:
4873:A playable example of the second puzzle
4812:
4808:
4806:
4804:
4544:there exist solutions for the equation
3308:is the function determining the check
2292:; it is put on the left balance pan if
1484:we shall denote the discrete -cube in
715:0, 3, 8, 10, 12 against 6, 7, 9, 11, 13
712:0, 2, 4, 10, 11 against 5, 8, 9, 12, 13
709:0, 1, 4, 5, 6 against 7, 10, 11, 12, 13
4887:
3472:of the algorithm from the results of
2055:{\displaystyle \mathrm {h} \in I^{n};}
555:
407:{\displaystyle {\frac {3^{n}-1}{2}}-1}
4815:Discrete Mathematics and Applications
4743:
2090:{\displaystyle \mathrm {x} \in I^{n}}
255:Target coin is different from others
4801:
513:
295:{\displaystyle {\frac {3^{n}-1}{2}}}
54:adding citations to reliable sources
25:
3655:{\displaystyle S(Z,{\mathcal {A}})}
2682:are called admissible situations.
13:
4718:Original parallel weighings puzzle
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3691:{\displaystyle (Z,{\mathcal {A}})}
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3570:weighings at the previous steps (
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1867:th object has standard weight if
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4650:there are no perfect WA, and for
2707:induces the partition of the set
2649:{\displaystyle Z\subseteq I^{n},}
969:{\displaystyle \mathbb {R} ^{n}.}
2436:{\displaystyle r(\mathrm {h} )=}
2014:{\displaystyle \mathrm {x} ^{+}}
1677:is the discrete ball of radius
1506:{\displaystyle \mathbb {R} ^{n}}
937:{\displaystyle \mathrm {e} ^{2}}
908:{\displaystyle \mathrm {e} ^{1}}
802:{\displaystyle \mathbb {R} ^{n}}
545:single light coin from a set of
30:
3465:{\displaystyle j=1,2,\dots ,m,}
2325:and is put on the right pan if
2179:The weighing given by a vector
1176:are defined, respectively, as
41:needs additional citations for
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4457:and in this sense is perfect.
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3973:{\displaystyle {\mathcal {A}}}
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3154:{\displaystyle {\mathcal {A}}}
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2761:
2745:
2601:
2585:
2553:
2537:
2508:
2492:
2469:
2453:
2430:
2404:
2398:
2390:
2226:
2194:
2163:
2160:
2144:
2141:
2123:
2107:
1818:
1786:
1771:objects are given by a vector
1713:
1710:
1642:
1634:
1627:
1432:
1416:
1374:
1369:
1365:
1352:
1336:
1332:
1288:
1272:
1230:
1226:
1213:
1198:
1157:
1150:
1124:
1117:
1023:
991:
870:
840:
453:
438:
341:
326:
235:
229:
1:
4730:
3137:. A weighing algorithm (WA)
1744:{\displaystyle \mathrm {0} .}
661:
564:
4330:{\displaystyle n=11,m=5,t=2}
2700:{\displaystyle \mathrm {h} }
2373:{\displaystyle \mathrm {h} }
1726:) with centre at the point
1573:{\displaystyle I=\{-1,0,1\}}
1169:{\displaystyle (\cdot )^{+}}
1136:{\displaystyle (\cdot )^{*}}
7:
4787:"Math Forum - Ask Dr. Math"
3620:is a given initial check).
2351:{\displaystyle h_{i}>0.}
2285:{\displaystyle h_{i}\neq 0}
535:
10:
4911:
2318:{\displaystyle h_{i}<0}
18:
4769:Grossman, Howard (1945).
4485:{\displaystyle I_{t}^{n}}
4707:{\displaystyle n=11,m=5}
1985:{\displaystyle x_{i}=-1}
1896:{\displaystyle x_{i}=0;}
885:inner product of vectors
728:Without a reference coin
162:Number of Weighings for
2914:{\displaystyle s\in I,}
1949:{\displaystyle x_{i}=1}
4708:
4670:
4644:
4620:(corresponding to the
4614:
4571:
4538:
4537:{\displaystyle t>2}
4512:
4486:
4451:
4421:
4331:
4279:
4256:
4234:
4195:is satisfied for all
4189:
4122:
4102:
4063:is satisfied for all
4057:
3994:
3974:
3945:
3862:
3759:
3739:
3692:
3656:
3614:
3564:
3466:
3419:
3399:
3302:
3241:
3175:
3155:
3126:
3034:
2915:
2886:
2774:
2728:
2701:
2676:
2675:{\displaystyle z\in Z}
2650:
2614:
2569:
2521:
2476:
2437:
2374:
2352:
2319:
2286:
2253:
2233:
2173:
2091:
2056:
2015:
1986:
1950:
1917:
1897:
1861:
1841:
1765:
1745:
1720:
1691:
1671:
1574:
1527:
1507:
1478:
1451:
1387:
1304:
1243:
1170:
1137:
1104:
1045:
970:
938:
909:
877:
823:
803:
524:
463:
408:
351:
296:
245:
199:
4827:10.1515/dma-2015-0007
4750:mathworld.Wolfram.com
4709:
4671:
4645:
4615:
4551:
4539:
4513:
4487:
4452:
4422:
4332:
4280:
4257:
4235:
4190:
4123:
4103:
4058:
3995:
3975:
3946:
3863:
3760:
3740:
3693:
3657:
3615:
3565:
3467:
3420:
3400:
3303:
3242:
3176:
3156:
3127:
3035:
2916:
2887:
2775:
2729:
2727:{\displaystyle I^{n}}
2702:
2677:
2651:
2615:
2570:
2522:
2477:
2438:
2375:
2353:
2320:
2287:
2254:
2234:
2174:
2092:
2057:
2016:
1987:
1951:
1918:
1898:
1862:
1842:
1766:
1751:Relative weights of
1746:
1721:
1692:
1672:
1575:
1528:
1508:
1479:
1477:{\displaystyle I^{n}}
1452:
1388:
1305:
1244:
1171:
1138:
1105:
1046:
971:
939:
910:
878:
824:
804:
701:With a reference coin
521:
464:
409:
352:
297:
246:
200:
198:{\displaystyle 3^{n}}
4680:
4654:
4628:
4548:
4522:
4496:
4464:
4435:
4349:
4291:
4266:
4246:
4199:
4132:
4112:
4067:
4004:
3984:
3960:
3872:
3769:
3749:
3702:
3666:
3627:
3574:
3476:
3429:
3409:
3312:
3251:
3185:
3165:
3141:
3044:
2925:
2896:
2784:
2742:
2711:
2689:
2660:
2624:
2579:
2531:
2486:
2447:
2384:
2362:
2329:
2296:
2263:
2243:
2183:
2101:
2066:
2028:
1996:
1960:
1927:
1907:
1871:
1851:
1775:
1755:
1730:
1704:
1681:
1584:
1537:
1517:
1488:
1461:
1397:
1314:
1253:
1180:
1147:
1114:
1055:
980:
948:
919:
890:
837:
813:
784:
419:
370:
307:
264:
210:
182:
50:improve this article
4771:Scripta Mathematica
4744:Weisstein, Eric W.
4669:{\displaystyle t=2}
4643:{\displaystyle t=1}
4596:
4511:{\displaystyle n,t}
4492:for some values of
4481:
4450:{\displaystyle t=2}
4403:
4375:
3662:be the set of all
1903:the weight of the
1719:{\displaystyle w()}
1601:
1533:over the alphabet
556:Twelve-coin problem
4704:
4666:
4640:
4610:
4582:
4534:
4508:
4482:
4467:
4447:
4417:
4389:
4361:
4339:ternary Golay code
4327:
4278:{\displaystyle Z.}
4275:
4252:
4230:
4185:
4128:if the condition
4118:
4098:
4053:
4000:if the condition
3990:
3970:
3941:
3858:
3755:
3735:
3688:
3652:
3610:
3560:
3462:
3415:
3395:
3298:
3237:
3171:
3151:
3122:
3030:
2911:
2882:
2780:into three parts
2773:{\displaystyle =0}
2770:
2724:
2697:
2672:
2646:
2610:
2565:
2517:
2472:
2433:
2370:
2358:For each weighing
2348:
2315:
2282:
2249:
2229:
2169:
2087:
2052:
2011:
1982:
1946:
1913:
1893:
1857:
1837:
1761:
1741:
1716:
1687:
1667:
1587:
1570:
1523:
1503:
1474:
1447:
1383:
1300:
1239:
1166:
1133:
1100:
1041:
966:
934:
905:
873:
819:
799:
525:
459:
404:
347:
292:
241:
195:
153:Maximum Coins for
4255:{\displaystyle Z}
4121:{\displaystyle Z}
3993:{\displaystyle Z}
3758:{\displaystyle s}
3418:{\displaystyle j}
3174:{\displaystyle m}
2252:{\displaystyle i}
1916:{\displaystyle i}
1860:{\displaystyle i}
1764:{\displaystyle n}
1690:{\displaystyle t}
1526:{\displaystyle n}
822:{\displaystyle n}
514:Nine-coin problem
472:
471:
396:
290:
126:
125:
118:
100:
4902:
4860:
4859:
4857:
4845:
4839:
4838:
4810:
4799:
4798:
4793:. Archived from
4783:
4777:
4767:
4761:
4760:
4758:
4756:
4741:
4713:
4711:
4710:
4705:
4675:
4673:
4672:
4667:
4649:
4647:
4646:
4641:
4619:
4617:
4616:
4611:
4609:
4608:
4595:
4590:
4581:
4580:
4570:
4565:
4543:
4541:
4540:
4535:
4517:
4515:
4514:
4509:
4491:
4489:
4488:
4483:
4480:
4475:
4456:
4454:
4453:
4448:
4426:
4424:
4423:
4418:
4416:
4415:
4402:
4397:
4388:
4387:
4374:
4369:
4336:
4334:
4333:
4328:
4284:
4282:
4281:
4276:
4261:
4259:
4258:
4253:
4239:
4237:
4236:
4231:
4223:
4222:
4194:
4192:
4191:
4186:
4178:
4170:
4169:
4160:
4149:
4148:
4139:
4127:
4125:
4124:
4119:
4107:
4105:
4104:
4099:
4091:
4090:
4062:
4060:
4059:
4054:
4046:
4038:
4037:
4025:
4011:
3999:
3997:
3996:
3991:
3979:
3977:
3976:
3971:
3969:
3968:
3950:
3948:
3947:
3942:
3928:
3927:
3921:
3901:
3900:
3888:
3867:
3865:
3864:
3859:
3845:
3844:
3838:
3824:
3819:
3818:
3806:
3792:
3791:
3785:
3764:
3762:
3761:
3756:
3744:
3742:
3741:
3736:
3725:
3724:
3718:
3698:-syndromes and
3697:
3695:
3694:
3689:
3684:
3683:
3661:
3659:
3658:
3653:
3648:
3647:
3619:
3617:
3616:
3611:
3603:
3602:
3597:
3588:
3587:
3582:
3569:
3567:
3566:
3561:
3559:
3558:
3537:
3536:
3512:
3511:
3496:
3495:
3484:
3471:
3469:
3468:
3463:
3424:
3422:
3421:
3416:
3404:
3402:
3401:
3396:
3391:
3390:
3378:
3377:
3372:
3360:
3359:
3341:
3340:
3335:
3326:
3325:
3320:
3307:
3305:
3304:
3299:
3297:
3296:
3284:
3283:
3265:
3264:
3259:
3246:
3244:
3243:
3238:
3230:
3229:
3224:
3209:
3208:
3203:
3194:
3193:
3180:
3178:
3177:
3172:
3160:
3158:
3157:
3152:
3150:
3149:
3131:
3129:
3128:
3123:
3109:
3101:
3100:
3091:
3071:
3060:
3039:
3037:
3036:
3031:
3023:
3012:
2989:
2978:
2958:
2947:
2920:
2918:
2917:
2912:
2891:
2889:
2888:
2883:
2869:
2861:
2850:
2845:
2844:
2832:
2818:
2810:
2809:
2800:
2779:
2777:
2776:
2771:
2760:
2752:
2733:
2731:
2730:
2725:
2723:
2722:
2706:
2704:
2703:
2698:
2696:
2681:
2679:
2678:
2673:
2655:
2653:
2652:
2647:
2642:
2641:
2619:
2617:
2616:
2611:
2600:
2592:
2574:
2572:
2571:
2566:
2552:
2544:
2526:
2524:
2523:
2518:
2507:
2499:
2481:
2479:
2478:
2473:
2468:
2460:
2442:
2440:
2439:
2434:
2411:
2397:
2379:
2377:
2376:
2371:
2369:
2357:
2355:
2354:
2349:
2341:
2340:
2324:
2322:
2321:
2316:
2308:
2307:
2291:
2289:
2288:
2283:
2275:
2274:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2225:
2224:
2206:
2205:
2190:
2178:
2176:
2175:
2170:
2159:
2151:
2122:
2114:
2096:
2094:
2093:
2088:
2086:
2085:
2073:
2061:
2059:
2058:
2053:
2048:
2047:
2035:
2020:
2018:
2017:
2012:
2010:
2009:
2004:
1991:
1989:
1988:
1983:
1972:
1971:
1955:
1953:
1952:
1947:
1939:
1938:
1922:
1920:
1919:
1914:
1902:
1900:
1899:
1894:
1883:
1882:
1866:
1864:
1863:
1858:
1846:
1844:
1843:
1838:
1833:
1832:
1817:
1816:
1798:
1797:
1782:
1770:
1768:
1767:
1762:
1750:
1748:
1747:
1742:
1737:
1725:
1723:
1722:
1717:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1666:
1665:
1641:
1630:
1625:
1624:
1612:
1600:
1595:
1579:
1577:
1576:
1571:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1502:
1501:
1496:
1483:
1481:
1480:
1475:
1473:
1472:
1456:
1454:
1453:
1448:
1440:
1439:
1430:
1429:
1424:
1409:
1408:
1392:
1390:
1389:
1384:
1382:
1381:
1372:
1364:
1363:
1339:
1328:
1327:
1322:
1309:
1307:
1306:
1301:
1296:
1295:
1286:
1285:
1280:
1265:
1264:
1248:
1246:
1245:
1240:
1238:
1237:
1225:
1224:
1194:
1193:
1188:
1175:
1173:
1172:
1167:
1165:
1164:
1142:
1140:
1139:
1134:
1132:
1131:
1110:the operations
1109:
1107:
1106:
1101:
1096:
1095:
1090:
1078:
1077:
1072:
1050:
1048:
1047:
1042:
1040:
1039:
1034:
1022:
1021:
1003:
1002:
987:
975:
973:
972:
967:
962:
961:
956:
943:
941:
940:
935:
933:
932:
927:
914:
912:
911:
906:
904:
903:
898:
882:
880:
879:
876:{\displaystyle }
874:
869:
868:
863:
854:
853:
848:
828:
826:
825:
820:
808:
806:
805:
800:
798:
797:
792:
548:
509:
507:
506:
503:
500:
489:
487:
486:
483:
480:
468:
466:
465:
460:
434:
433:
413:
411:
410:
405:
397:
392:
385:
384:
374:
356:
354:
353:
348:
322:
321:
301:
299:
298:
293:
291:
286:
279:
278:
268:
250:
248:
247:
242:
225:
224:
204:
202:
201:
196:
194:
193:
167:
158:
144:
143:
121:
114:
110:
107:
101:
99:
65:"Balance puzzle"
58:
34:
26:
21:Balance (puzzle)
4910:
4909:
4905:
4904:
4903:
4901:
4900:
4899:
4885:
4884:
4869:
4864:
4863:
4846:
4842:
4811:
4802:
4785:
4784:
4780:
4768:
4764:
4754:
4752:
4742:
4738:
4733:
4720:
4681:
4678:
4677:
4655:
4652:
4651:
4629:
4626:
4625:
4604:
4600:
4591:
4586:
4576:
4572:
4566:
4555:
4549:
4546:
4545:
4523:
4520:
4519:
4497:
4494:
4493:
4476:
4471:
4465:
4462:
4461:
4436:
4433:
4432:
4411:
4407:
4398:
4393:
4383:
4379:
4370:
4365:
4350:
4347:
4346:
4292:
4289:
4288:
4267:
4264:
4263:
4247:
4244:
4243:
4218:
4217:
4200:
4197:
4196:
4174:
4165:
4164:
4156:
4144:
4140:
4135:
4133:
4130:
4129:
4113:
4110:
4109:
4086:
4085:
4068:
4065:
4064:
4042:
4033:
4032:
4021:
4007:
4005:
4002:
4001:
3985:
3982:
3981:
3964:
3963:
3961:
3958:
3957:
3923:
3922:
3917:
3896:
3895:
3884:
3873:
3870:
3869:
3840:
3839:
3834:
3820:
3814:
3810:
3802:
3787:
3786:
3781:
3770:
3767:
3766:
3750:
3747:
3746:
3720:
3719:
3714:
3703:
3700:
3699:
3679:
3678:
3667:
3664:
3663:
3643:
3642:
3628:
3625:
3624:
3598:
3593:
3592:
3583:
3578:
3577:
3575:
3572:
3571:
3548:
3544:
3526:
3522:
3507:
3503:
3485:
3480:
3479:
3477:
3474:
3473:
3430:
3427:
3426:
3410:
3407:
3406:
3386:
3382:
3373:
3368:
3367:
3349:
3345:
3336:
3331:
3330:
3321:
3316:
3315:
3313:
3310:
3309:
3292:
3288:
3273:
3269:
3260:
3255:
3254:
3252:
3249:
3248:
3225:
3220:
3219:
3204:
3199:
3198:
3189:
3188:
3186:
3183:
3182:
3181:is a sequence
3166:
3163:
3162:
3145:
3144:
3142:
3139:
3138:
3105:
3096:
3092:
3087:
3067:
3056:
3045:
3042:
3041:
3019:
3008:
2985:
2974:
2954:
2943:
2926:
2923:
2922:
2897:
2894:
2893:
2865:
2857:
2846:
2840:
2836:
2828:
2814:
2805:
2801:
2796:
2785:
2782:
2781:
2756:
2748:
2743:
2740:
2739:
2718:
2714:
2712:
2709:
2708:
2692:
2690:
2687:
2686:
2661:
2658:
2657:
2637:
2633:
2625:
2622:
2621:
2596:
2588:
2580:
2577:
2576:
2548:
2540:
2532:
2529:
2528:
2503:
2495:
2487:
2484:
2483:
2464:
2456:
2448:
2445:
2444:
2407:
2393:
2385:
2382:
2381:
2365:
2363:
2360:
2359:
2336:
2332:
2330:
2327:
2326:
2303:
2299:
2297:
2294:
2293:
2270:
2266:
2264:
2261:
2260:
2244:
2241:
2240:
2220:
2216:
2201:
2197:
2186:
2184:
2181:
2180:
2155:
2147:
2118:
2110:
2102:
2099:
2098:
2081:
2077:
2069:
2067:
2064:
2063:
2043:
2039:
2031:
2029:
2026:
2025:
2005:
2000:
1999:
1997:
1994:
1993:
1992:). The vector
1967:
1963:
1961:
1958:
1957:
1956:(respectively,
1934:
1930:
1928:
1925:
1924:
1908:
1905:
1904:
1878:
1874:
1872:
1869:
1868:
1852:
1849:
1848:
1828:
1824:
1812:
1808:
1793:
1789:
1778:
1776:
1773:
1772:
1756:
1753:
1752:
1733:
1731:
1728:
1727:
1705:
1702:
1701:
1682:
1679:
1678:
1661:
1657:
1637:
1626:
1620:
1616:
1608:
1596:
1591:
1585:
1582:
1581:
1538:
1535:
1534:
1518:
1515:
1514:
1497:
1492:
1491:
1489:
1486:
1485:
1468:
1464:
1462:
1459:
1458:
1435:
1431:
1425:
1420:
1419:
1404:
1400:
1398:
1395:
1394:
1377:
1373:
1368:
1359:
1355:
1335:
1323:
1318:
1317:
1315:
1312:
1311:
1291:
1287:
1281:
1276:
1275:
1260:
1256:
1254:
1251:
1250:
1233:
1229:
1220:
1216:
1189:
1184:
1183:
1181:
1178:
1177:
1160:
1156:
1148:
1145:
1144:
1127:
1123:
1115:
1112:
1111:
1091:
1086:
1085:
1073:
1068:
1067:
1056:
1053:
1052:
1035:
1030:
1029:
1017:
1013:
998:
994:
983:
981:
978:
977:
957:
952:
951:
949:
946:
945:
928:
923:
922:
920:
917:
916:
899:
894:
893:
891:
888:
887:
864:
859:
858:
849:
844:
843:
838:
835:
834:
831:Euclidean space
814:
811:
810:
793:
788:
787:
785:
782:
781:
775:
773:Generalizations
766:
759:
752:
730:
703:
664:
567:
558:
546:
538:
516:
504:
501:
498:
497:
495:
484:
481:
478:
477:
475:
429:
425:
420:
417:
416:
380:
376:
375:
373:
371:
368:
367:
317:
313:
308:
305:
304:
274:
270:
269:
267:
265:
262:
261:
220:
216:
211:
208:
207:
189:
185:
183:
180:
179:
163:
154:
134:weighing puzzle
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
4908:
4898:
4897:
4883:
4882:
4875:
4868:
4867:External links
4865:
4862:
4861:
4840:
4800:
4797:on 2002-06-12.
4778:
4762:
4735:
4734:
4732:
4729:
4719:
4716:
4703:
4700:
4697:
4694:
4691:
4688:
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4665:
4662:
4659:
4639:
4636:
4633:
4607:
4603:
4599:
4594:
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4585:
4579:
4575:
4569:
4564:
4561:
4558:
4554:
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4530:
4527:
4507:
4504:
4501:
4479:
4474:
4470:
4446:
4443:
4440:
4414:
4410:
4406:
4401:
4396:
4392:
4386:
4382:
4378:
4373:
4368:
4364:
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4357:
4354:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4274:
4271:
4251:
4229:
4226:
4221:
4216:
4213:
4210:
4207:
4204:
4184:
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4177:
4173:
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4159:
4155:
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4138:
4117:
4097:
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4084:
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4078:
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4017:
4014:
4010:
3989:
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3848:
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3827:
3823:
3817:
3813:
3809:
3805:
3801:
3798:
3795:
3790:
3784:
3780:
3777:
3774:
3754:
3734:
3731:
3728:
3723:
3717:
3713:
3710:
3707:
3687:
3682:
3677:
3674:
3671:
3651:
3646:
3641:
3638:
3635:
3632:
3609:
3606:
3601:
3596:
3591:
3586:
3581:
3557:
3554:
3551:
3547:
3543:
3540:
3535:
3532:
3529:
3525:
3521:
3518:
3515:
3510:
3506:
3502:
3499:
3494:
3491:
3488:
3483:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3414:
3394:
3389:
3385:
3381:
3376:
3371:
3366:
3363:
3358:
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3348:
3344:
3339:
3334:
3329:
3324:
3319:
3295:
3291:
3287:
3282:
3279:
3276:
3272:
3268:
3263:
3258:
3236:
3233:
3228:
3223:
3218:
3215:
3212:
3207:
3202:
3197:
3192:
3170:
3148:
3121:
3118:
3115:
3112:
3108:
3104:
3099:
3095:
3090:
3086:
3083:
3080:
3077:
3074:
3070:
3066:
3063:
3059:
3055:
3052:
3049:
3029:
3026:
3022:
3018:
3015:
3011:
3007:
3004:
3001:
2998:
2995:
2992:
2988:
2984:
2981:
2977:
2973:
2970:
2967:
2964:
2961:
2957:
2953:
2950:
2946:
2942:
2939:
2936:
2933:
2930:
2910:
2907:
2904:
2901:
2881:
2878:
2875:
2872:
2868:
2864:
2860:
2856:
2853:
2849:
2843:
2839:
2835:
2831:
2827:
2824:
2821:
2817:
2813:
2808:
2804:
2799:
2795:
2792:
2789:
2769:
2766:
2763:
2759:
2755:
2751:
2747:
2734:by the plane (
2721:
2717:
2695:
2685:Each weighing
2671:
2668:
2665:
2645:
2640:
2636:
2632:
2629:
2609:
2606:
2603:
2599:
2595:
2591:
2587:
2584:
2564:
2561:
2558:
2555:
2551:
2547:
2543:
2539:
2536:
2516:
2513:
2510:
2506:
2502:
2498:
2494:
2491:
2471:
2467:
2463:
2459:
2455:
2452:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2410:
2406:
2403:
2400:
2396:
2392:
2389:
2368:
2347:
2344:
2339:
2335:
2314:
2311:
2306:
2302:
2281:
2278:
2273:
2269:
2248:
2228:
2223:
2219:
2215:
2212:
2209:
2204:
2200:
2196:
2193:
2189:
2168:
2165:
2162:
2158:
2154:
2150:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2121:
2117:
2113:
2109:
2106:
2084:
2080:
2076:
2072:
2051:
2046:
2042:
2038:
2034:
2008:
2003:
1981:
1978:
1975:
1970:
1966:
1945:
1942:
1937:
1933:
1912:
1892:
1889:
1886:
1881:
1877:
1856:
1836:
1831:
1827:
1823:
1820:
1815:
1811:
1807:
1804:
1801:
1796:
1792:
1788:
1785:
1781:
1760:
1740:
1736:
1715:
1712:
1709:
1699:Hamming metric
1686:
1664:
1660:
1656:
1653:
1650:
1647:
1644:
1640:
1636:
1633:
1629:
1623:
1619:
1615:
1611:
1607:
1604:
1599:
1594:
1590:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1522:
1500:
1495:
1471:
1467:
1446:
1443:
1438:
1434:
1428:
1423:
1418:
1415:
1412:
1407:
1403:
1380:
1376:
1371:
1367:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1338:
1334:
1331:
1326:
1321:
1299:
1294:
1290:
1284:
1279:
1274:
1271:
1268:
1263:
1259:
1236:
1232:
1228:
1223:
1219:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1192:
1187:
1163:
1159:
1155:
1152:
1130:
1126:
1122:
1119:
1099:
1094:
1089:
1084:
1081:
1076:
1071:
1066:
1063:
1060:
1038:
1033:
1028:
1025:
1020:
1016:
1012:
1009:
1006:
1001:
997:
993:
990:
986:
965:
960:
955:
931:
926:
902:
897:
872:
867:
862:
857:
852:
847:
842:
818:
796:
791:
774:
771:
764:
757:
750:
745:
744:
741:
729:
726:
717:
716:
713:
710:
702:
699:
698:
697:
696:
695:
691:
684:
683:
682:
679:
672:
671:
663:
660:
659:
658:
655:
654:
653:
652:
646:
645:
644:
643:
636:
635:
634:
633:
626:
625:
624:possibilities:
620:
619:
618:
617:
610:
609:
608:
607:
600:
599:
598:
597:
590:
589:
588:possibilities:
584:
583:
566:
563:
557:
554:
547:3 × 3 = 9
537:
534:
515:
512:
470:
469:
458:
455:
452:
449:
446:
443:
440:
437:
432:
428:
424:
414:
403:
400:
395:
391:
388:
383:
379:
365:
362:
358:
357:
346:
343:
340:
337:
334:
331:
328:
325:
320:
316:
312:
302:
289:
285:
282:
277:
273:
259:
258:Identify coin
256:
252:
251:
240:
237:
234:
231:
228:
223:
219:
215:
205:
192:
188:
177:
176:Identify coin
174:
170:
169:
160:
151:
148:
130:balance puzzle
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
4907:
4896:
4895:Logic puzzles
4893:
4892:
4890:
4881:
4880:
4876:
4874:
4871:
4870:
4856:
4851:
4844:
4836:
4832:
4828:
4824:
4820:
4816:
4809:
4807:
4805:
4796:
4792:
4791:mathforum.org
4788:
4782:
4775:
4772:
4766:
4751:
4747:
4740:
4736:
4728:
4725:
4715:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4663:
4660:
4657:
4637:
4634:
4631:
4623:
4622:Hamming bound
4605:
4601:
4597:
4592:
4587:
4583:
4577:
4573:
4567:
4562:
4559:
4556:
4552:
4531:
4528:
4525:
4505:
4502:
4499:
4477:
4472:
4468:
4458:
4444:
4441:
4438:
4430:
4429:Hamming bound
4412:
4408:
4404:
4399:
4394:
4390:
4384:
4380:
4376:
4371:
4366:
4362:
4358:
4355:
4352:
4342:
4340:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4285:
4272:
4269:
4249:
4240:
4227:
4214:
4208:
4205:
4202:
4182:
4179:
4161:
4153:
4145:
4141:
4115:
4095:
4082:
4076:
4073:
4070:
4050:
4047:
4029:
4026:
4018:
4012:
3987:
3955:
3951:
3938:
3935:
3932:
3914:
3908:
3905:
3892:
3889:
3881:
3875:
3852:
3849:
3831:
3825:
3815:
3811:
3807:
3796:
3778:
3772:
3752:
3732:
3729:
3711:
3705:
3675:
3672:
3639:
3636:
3630:
3621:
3599:
3589:
3584:
3555:
3552:
3549:
3545:
3541:
3533:
3530:
3527:
3523:
3519:
3516:
3513:
3508:
3504:
3497:
3492:
3489:
3486:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3412:
3392:
3387:
3383:
3379:
3374:
3364:
3356:
3353:
3350:
3346:
3337:
3327:
3322:
3293:
3289:
3280:
3277:
3274:
3270:
3266:
3261:
3234:
3231:
3226:
3216:
3213:
3210:
3205:
3195:
3168:
3136:
3132:
3119:
3116:
3113:
3102:
3097:
3093:
3084:
3078:
3075:
3064:
3061:
3053:
3047:
3027:
3016:
3013:
3005:
3002:
2996:
2993:
2982:
2979:
2971:
2965:
2962:
2951:
2948:
2940:
2934:
2931:
2928:
2908:
2905:
2902:
2899:
2876:
2873:
2862:
2851:
2841:
2837:
2833:
2822:
2811:
2806:
2802:
2793:
2787:
2767:
2764:
2753:
2737:
2719:
2715:
2683:
2669:
2666:
2663:
2643:
2638:
2634:
2630:
2627:
2607:
2604:
2593:
2582:
2562:
2559:
2556:
2545:
2534:
2514:
2511:
2500:
2489:
2461:
2450:
2427:
2424:
2421:
2418:
2415:
2412:
2401:
2387:
2345:
2342:
2337:
2333:
2312:
2309:
2304:
2300:
2279:
2276:
2271:
2267:
2246:
2221:
2217:
2213:
2210:
2207:
2202:
2198:
2191:
2166:
2152:
2138:
2135:
2132:
2129:
2126:
2115:
2104:
2082:
2078:
2074:
2049:
2044:
2040:
2036:
2022:
2006:
1979:
1976:
1973:
1968:
1964:
1943:
1940:
1935:
1931:
1910:
1890:
1887:
1884:
1879:
1875:
1854:
1834:
1829:
1825:
1821:
1813:
1809:
1805:
1802:
1799:
1794:
1790:
1783:
1758:
1738:
1734:
1707:
1700:
1684:
1662:
1658:
1654:
1648:
1645:
1631:
1621:
1617:
1613:
1602:
1597:
1592:
1588:
1564:
1561:
1558:
1555:
1552:
1549:
1543:
1540:
1520:
1498:
1469:
1465:
1444:
1436:
1426:
1410:
1405:
1401:
1378:
1360:
1356:
1349:
1346:
1343:
1340:
1329:
1324:
1292:
1282:
1266:
1261:
1257:
1234:
1221:
1217:
1210:
1207:
1204:
1201:
1195:
1190:
1161:
1153:
1128:
1120:
1097:
1092:
1082:
1074:
1061:
1058:
1051:and subsets
1036:
1026:
1018:
1014:
1010:
1007:
1004:
999:
995:
988:
976:For vectors
963:
958:
929:
900:
886:
865:
855:
850:
832:
829:-dimensional
816:
794:
778:
770:
763:
756:
749:
742:
739:
738:
737:
734:
725:
721:
714:
711:
708:
707:
706:
692:
688:
687:
685:
680:
677:
676:
674:
673:
669:
668:
667:
657:
656:
650:
649:
648:
647:
640:
639:
638:
637:
630:
629:
628:
627:
622:
621:
614:
613:
612:
611:
604:
603:
602:
601:
594:
593:
592:
591:
586:
585:
581:
580:
579:
577:
573:
562:
553:
550:
542:
533:
531:
520:
511:
493:
450:
447:
444:
441:
435:
430:
426:
415:
401:
398:
393:
389:
386:
381:
377:
366:
363:
360:
359:
338:
335:
332:
329:
323:
318:
314:
303:
287:
283:
280:
275:
271:
260:
257:
254:
253:
232:
226:
221:
217:
206:
190:
186:
178:
175:
172:
171:
166:
161:
157:
152:
149:
146:
145:
142:
139:
135:
131:
120:
117:
109:
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
4878:
4843:
4821:(2): 69–81.
4818:
4814:
4795:the original
4790:
4781:
4773:
4770:
4765:
4753:. Retrieved
4749:
4739:
4723:
4721:
4459:
4343:
4286:
4241:
3954:Definition 2
3953:
3952:
3622:
3135:Definition 1
3134:
3133:
2684:
2023:
779:
776:
767:
760:
755:irrelevant:
753:
746:
735:
731:
722:
718:
704:
665:
575:
571:
568:
559:
551:
543:
539:
526:
491:
473:
164:
155:
138:logic puzzle
133:
129:
127:
112:
106:January 2014
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
16:Logic puzzle
3161:of length
4746:"Weighing"
4731:References
2736:hyperplane
662:Variations
159:weighings
76:newspapers
4855:1310.7268
4835:124796871
4755:16 August
4553:∑
4206:∈
4074:∈
3933:∩
3808:∈
3730:⊆
3553:−
3542:∈
3531:−
3517:…
3490:−
3451:…
3425:th step,
3405:at each
3380:∈
3354:−
3286:→
3278:−
3214:…
3114:∩
3003:−
2903:∈
2834:∈
2667:∈
2631:⊆
2560:−
2422:…
2277:≠
2211:…
2075:∈
2037:∈
1977:−
1822:∈
1803:…
1655:⊆
1646:≤
1614:∈
1580:The set
1550:−
1293:∗
1262:∗
1191:∗
1154:⋅
1129:∗
1121:⋅
1083:⊆
1027:∈
1008:…
457:⌉
436:
423:⌈
399:−
387:−
345:⌉
324:
311:⌈
281:−
239:⌉
227:
214:⌈
4889:Category
3956:. A WA
3765:; i.e.,
1697:(in the
1249: ;
565:Solution
536:Solution
3247:where
3040:where
883:be the
809:be the
596:puzzle.
523:swapped
508:
496:
488:
476:
90:scholar
4833:
3623:Let
944:from
168:coins
147:Known
92:
85:
78:
71:
63:
4850:arXiv
4831:S2CID
3196:=<
1143:and
915:and
690:coin.
530:scale
499:3 − 1
479:3 − 1
150:Goal
136:is a
97:JSTOR
83:books
4757:2017
4529:>
4431:for
3232:>
2343:>
2310:<
1457:By
833:and
780:Let
616:one.
69:news
4823:doi
3868:;
2892:,
2097:is
427:log
315:log
218:log
132:or
52:by
4891::
4829:.
4819:25
4817:.
4803:^
4789:.
4774:XI
4748:.
4690:11
4395:11
4367:11
4301:11
2738:)
2608:1.
2346:0.
1393:,
1310:,
549:.
128:A
4858:.
4852::
4837:.
4825::
4776:.
4759:.
4724:N
4702:5
4699:=
4696:m
4693:,
4687:=
4684:n
4664:2
4661:=
4658:t
4638:1
4635:=
4632:t
4606:m
4602:3
4598:=
4593:i
4588:n
4584:C
4578:i
4574:2
4568:t
4563:0
4560:=
4557:i
4532:2
4526:t
4506:t
4503:,
4500:n
4478:n
4473:t
4469:I
4445:2
4442:=
4439:t
4413:5
4409:3
4405:=
4400:2
4391:C
4385:2
4381:2
4377:+
4372:1
4363:C
4359:2
4356:+
4353:1
4325:2
4322:=
4319:t
4316:,
4313:5
4310:=
4307:m
4304:,
4298:=
4295:n
4273:.
4270:Z
4250:Z
4228:.
4225:)
4220:A
4215:Z
4212:(
4209:S
4203:s
4183:1
4180:=
4176:|
4172:)
4167:A
4162:Z
4158:|
4154:s
4151:(
4146:+
4142:W
4137:|
4116:Z
4096:;
4093:)
4088:A
4083:Z
4080:(
4077:S
4071:s
4051:1
4048:=
4044:|
4040:)
4035:A
4030:,
4027:Z
4023:|
4019:s
4016:(
4013:W
4009:|
3988:Z
3966:A
3939:.
3936:Z
3930:)
3925:A
3919:|
3915:s
3912:(
3909:W
3906:=
3903:)
3898:A
3893:;
3890:Z
3886:|
3882:s
3879:(
3876:W
3856:}
3853:s
3850:=
3847:)
3842:A
3836:|
3832:z
3829:(
3826:s
3822:|
3816:m
3812:I
3804:z
3800:{
3797:=
3794:)
3789:A
3783:|
3779:s
3776:(
3773:W
3753:s
3733:I
3727:)
3722:A
3716:|
3712:s
3709:(
3706:W
3686:)
3681:A
3676:,
3673:Z
3670:(
3650:)
3645:A
3640:,
3637:Z
3634:(
3631:S
3608:)
3605:(
3600:1
3595:A
3590:=
3585:1
3580:h
3556:1
3550:j
3546:I
3539:)
3534:1
3528:j
3524:s
3520:,
3514:,
3509:1
3505:s
3501:(
3498:=
3493:1
3487:j
3482:s
3460:,
3457:m
3454:,
3448:,
3445:2
3442:,
3439:1
3436:=
3433:j
3413:j
3393:,
3388:n
3384:I
3375:j
3370:h
3365:;
3362:)
3357:1
3351:j
3347:s
3343:(
3338:j
3333:A
3328:=
3323:j
3318:h
3294:n
3290:I
3281:1
3275:j
3271:I
3267::
3262:j
3257:A
3235:,
3227:m
3222:A
3217:,
3211:,
3206:1
3201:A
3191:A
3169:m
3147:A
3120:.
3117:Z
3111:)
3107:h
3103:,
3098:n
3094:I
3089:|
3085:s
3082:(
3079:W
3076:=
3073:)
3069:h
3065:,
3062:Z
3058:|
3054:s
3051:(
3048:W
3028:,
3025:)
3021:h
3017:,
3014:Z
3010:|
3006:1
3000:(
2997:W
2994:+
2991:)
2987:h
2983:,
2980:Z
2976:|
2972:1
2969:(
2966:W
2963:+
2960:)
2956:h
2952:,
2949:Z
2945:|
2941:0
2938:(
2935:W
2932:=
2929:Z
2909:,
2906:I
2900:s
2880:}
2877:s
2874:=
2871:)
2867:h
2863:;
2859:x
2855:(
2852:s
2848:|
2842:n
2838:I
2830:x
2826:{
2823:=
2820:)
2816:h
2812:;
2807:n
2803:I
2798:|
2794:s
2791:(
2788:W
2768:0
2765:=
2762:]
2758:h
2754:;
2750:x
2746:[
2720:n
2716:I
2694:h
2670:Z
2664:z
2644:,
2639:n
2635:I
2628:Z
2605:=
2602:)
2598:h
2594:;
2590:x
2586:(
2583:s
2563:1
2557:=
2554:)
2550:h
2546:;
2542:x
2538:(
2535:s
2515:0
2512:=
2509:)
2505:h
2501:;
2497:x
2493:(
2490:s
2470:)
2466:h
2462:;
2458:x
2454:(
2451:s
2431:]
2428:1
2425:,
2419:,
2416:1
2413:;
2409:h
2405:[
2402:=
2399:)
2395:h
2391:(
2388:r
2367:h
2338:i
2334:h
2313:0
2305:i
2301:h
2280:0
2272:i
2268:h
2247:i
2227:)
2222:n
2218:h
2214:,
2208:,
2203:1
2199:h
2195:(
2192:=
2188:h
2167:.
2164:)
2161:]
2157:h
2153:;
2149:x
2145:[
2142:(
2139:n
2136:g
2133:i
2130:s
2127:=
2124:)
2120:h
2116:;
2112:x
2108:(
2105:s
2083:n
2079:I
2071:x
2050:;
2045:n
2041:I
2033:h
2007:+
2002:x
1980:1
1974:=
1969:i
1965:x
1944:1
1941:=
1936:i
1932:x
1911:i
1891:;
1888:0
1885:=
1880:i
1876:x
1855:i
1835:,
1830:n
1826:I
1819:)
1814:n
1810:x
1806:,
1800:,
1795:1
1791:x
1787:(
1784:=
1780:x
1759:n
1739:.
1735:0
1714:)
1711:(
1708:w
1685:t
1663:n
1659:I
1652:}
1649:t
1643:)
1639:x
1635:(
1632:w
1628:|
1622:n
1618:I
1610:x
1606:{
1603:=
1598:n
1593:t
1589:I
1568:}
1565:1
1562:,
1559:0
1556:,
1553:1
1547:{
1544:=
1541:I
1521:n
1499:n
1494:R
1470:n
1466:I
1445:.
1442:}
1437:+
1433:)
1427:j
1422:e
1417:(
1414:{
1411:=
1406:+
1402:E
1379:i
1375:)
1370:|
1366:)
1361:i
1357:e
1353:(
1350:n
1347:g
1344:i
1341:s
1337:|
1333:(
1330:=
1325:+
1320:e
1298:}
1289:)
1283:j
1278:e
1273:(
1270:{
1267:=
1258:E
1235:i
1231:)
1227:)
1222:i
1218:e
1214:(
1211:n
1208:g
1205:i
1202:s
1199:(
1196:=
1186:e
1162:+
1158:)
1151:(
1125:)
1118:(
1098:,
1093:n
1088:R
1080:}
1075:j
1070:e
1065:{
1062:=
1059:E
1037:n
1032:R
1024:)
1019:n
1015:e
1011:,
1005:,
1000:1
996:e
992:(
989:=
985:e
964:.
959:n
954:R
930:2
925:e
901:1
896:e
871:]
866:2
861:e
856:,
851:1
846:e
841:[
817:n
795:n
790:R
576:n
572:n
505:2
502:/
492:n
485:2
482:/
454:)
451:3
448:+
445:c
442:2
439:(
431:3
402:1
394:2
390:1
382:n
378:3
342:)
339:1
336:+
333:c
330:2
327:(
319:3
288:2
284:1
276:n
272:3
236:)
233:c
230:(
222:3
191:n
187:3
165:c
156:n
119:)
113:(
108:)
104:(
94:·
87:·
80:·
73:·
46:.
23:.
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