For the purposes of the exploration, volunteers will be labeled, Spectator A, Spectator B, Spectator C, and Spectator D. Throughout this exploration, the coins used will be in CAD. The magician brings $1.96 to a table, consisting of six pennies, six nickels, six dimes, and four quarters. Spectator A is asked to pick up one coin. Spectator B is asked to pick up a different valued coin. This process is repeated for Spectator C and Spectator D. Then, the performer asks Spectator D to pick up coins which adds up to four times the value of the original coin taken. Spectator A is asked to pick up the same value of coins they picked up before. Spectator B is asked to pick up double the value of coins they picked up, and Spectator C is asked to pick up three times the initial value of the coin they were asked to pick up. The magician then turns around and correctly identifies the initial value of the coin picked up by each of the spectators.
Consider the following example:
Spectator Coin Initially Chosen Final Value of Coins
A Penny $0.01 + $0.01 = $0.02
B Nickel $0.05 + 2($0.05) = $0.15
C Dime $0.10 + 3($0.10) = $0.40
D Quarter $0.25 + 4(0.25) = $1.25
Let T ≡ the total value of coins left on the table.
In the example above:
T = $1.96 - $1.82
= $0.14 See if this can be put on one line, disregarding the $1.96 - $1.82.
During the first phase of the trick, where the spectators are asked to pick one coin, that is different from every other spectator, T = $1.96 – ($0.01 + $0.05 + $0.10 + $0.25), where T = $1.55. If I let p ≡ total number of pennies, n ≡ total number of nickels, d ≡ total number of dimes, and q ≡ total number of quarters, then, after the second phase, the total value of coins left on the table is:
T = 1.55 – (0.01p + 0.05n + 0.10d + 0.25q)
The equation above can also be written in terms of cents, rather than dollars, such that the decimals can be removed:
T = 155 – (p + 5n + 10d + 25q)
At this point in my exploration, I was stuck as I was unable to see the correlation between using the formula above, and the coin the spectator chose. The formula above was able to provide...