Batch 3 - Class 150 - Ternary Counting Problems

Homework:

In the problem with 4 weights, what if you can put the weights on either of the balances? How many different measures can you weigh now?

What system of counting does this look like - how many values can each weight take?

-1, 0 or 1 depending on whether you are keeping it on left side, skipping it, or keeping it on right
3 values per weight -hence ternary system
What are the weights we should use in ternary system - same as place values of ternary system (1,3,9,27)
So how many measurements can we support - 1+3+9+27=40

Attendance Liza, Damini, Siddhant, Zorawar, Arjun, Manya, Muskaan, Vir

Class puzzles

Recap of Binary System

Lets get to binary counting - base 2. What digits can we use and what are the place values?
Exercise 1: Ask kids to write certain decimal numbers in binary
Exercise 2: Ask kids to read certain binary numbers in decimal
Exercise 3: Counting - Count using five fingers. Revise how each digit moves (first one toggles, and each one then toggles when the one of its left goes to 0)
How can computers store binary values? Charge, magnetic field...
How can we send a binary number to someone?

Coin weighing

There are 3 coins, of which all are the same weight except one which is heavier. How many weighings on balance do you need to figure out the heavier coin?
There are 9 coins, of which all are the same weight except one which is heavier. Can you figure out in two weighings of balance which one is the heavier coin?

See if kids can solve it just through logical thinking
See if they can get to groups of 3 - why 3? The balance has three outcomes - heavier, lighter, same)
Map heavier/ lighter/ same as ternary digits 0, 1, 2. Can you now write down all the outcomes as ternary numbers? How many such ternary numbers exist?
Can you do the inverse - if we label the coins 0..8, write their ternary notation and figure out where each coin should go in the two weighings

See if kids can come up with the answer on their own
How many total outcomes are there? How many weighing outcomes can you get using 2 weighings? Can you map these?
May be use -1, 0, 1 as the ternary digits (or you can use 0, 1, 2 and see what happens if a particular coin is heavy or light)
Note: Kids correctly proved that this problem is not solvable as stated. If one has spare standard coins, then it is doable in 2 weighings.

Guess the card

Ternary system: http://www.geometer.org/mathcircles/CardTricks.pdf - Problem 1, Base 3 math through card trick - 27 cards in three columns, every time the subject tells which column the card is in. You assemble that column in middle of other two columns. Now spread the cards again row wise into three columns. At end of three assemblies, chosen card is the center card.

Homework Problem:

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:

References:

NEXT CLASS CAN BE ON BASE ARITHMETICS

NOT USED: http://www.mathteacherscircle.org/assets/session-materials/JTantonDots_Boxes_Piles_Holes_MiddleSchoolNotes.pdf - another way to explain binary, ternary and decimals numbers and do basic arithmetic operations on those