Batch 3 - Class 151 - Weighing coins and balls

• There are 12 coins, all identical in appearance, and all identical in weight except for one, which is either heavier or lighter than the remaining 11 coins. Devise a procedure to identify the counterfeit coin in only 3 weighings with a balance.
• Answer: This can be thought of as a ternary problem. With three weighings, we can get to 27 outcomes (on each weighing, the balance is either heavy on left, balanced, or heavy on right). With 12 coins, we can have 24 outcomes (coin number 1..12 being defective, with possibility of it being heavy or light). We can arrange the problem so that the ternary value of the weighings corresponds to defective coin, and by using digits -1, 0, 1 the actual coin can be identified with the number it forms. So for example, if coin 6 is heavy, then its 1, -1, 0 and hence coin 6 should appear in left pan on first weighing, right pan on second weighing and be absent on third. Notice that we will get -6 automatically if coin 6 was light. Coins 7,9,10,11 are flipped to balance the number of coins in each weighing, so they for 7 the code is (-1,1,-1) corresponding to what would be -7 normally. This will give us the following answer:
• (9) first weighing                            5,6,8,10                    7,9,11,12
• (3) second weighing                       2,3,4,7                      5,6,11,12
• (1) third weighing                          1,4,10,11                   2,5,7,8 
• See https://www.cut-the-knot.org/blue/weight1.shtml

• There are 10 sacks of balls. All balls are genuine and weight 10 gm, except one sack in which all the balls are lighter or heavier by 1 gm. Can you figure out the defective sack with one weighing? You have a single-pan scale, which tell you the total weight of what you put on the pan.
• Answer: Take one ball from first sack, two from second, three from third and so on. Weigh them - the difference from 550 tells you the sack number that is defective.

• There are four coins of which one is counterfeit. There is a fifth coin known to be good. Find the counterfeit coin and heavier/lighter with two weighings on a balance scale with two pans.
• Answer: https://www.cut-the-knot.org/blue/5Coins1Good1Bad.shtml#solution

• There are six balls - two red, two blue, two orange. One of each color is lighter (all balls that are lighter are the same weight, all balls that are heavier are of the same weight). Use two weighings to determine which balls are heavy or light
• Answer: See https://www.cut-the-knot.org/blue/6Balls2Weighings.shtml#solution

• Of 101 coins, 50 are counterfeit. All genuine coins weigh the same, all counterfeit couis weight the same and differ from the genuine ones by 1gm. You have a two-pan pointer scale which shows the difference in weights between the objects places in two pans. Can you take one coin and determine whether it is fake or genuine by one weighing?
• Answer: https://www.cut-the-knot.org/blue/OneWeighing.shtml#solution

• Find the missing number in the sequence 10000, ?, 100, 31, 24, 22, 20, 17, 16, 15, 14, 13, 12, 11, 10
• Answer: This is number 16 written in base 2 to 16. The missing number is 16 written in base 3, i.e. 121