High Rollers: Game Show and Possible Sums with the TI-84
Plus
If you have a
group of numbers, what are the possible sums you can make? We’re talking about combinations of 1, 2, to
as many numbers you have. This is one of
the key elements of the classic game show High Rollers (1970s, 1987-1988), a
dice rolling game based on Shut The Box.
To whet your
appetite, click on the links below to see classic episodes (the links should be
good as of 3/15/2017). The first two
links are from episodes from the 1978-1980 series with Alec Trebek (with facial
hair) and the last two links are from episodes from the 1987-1988 series with
Wink Martindale. We’ll get to the game show later in this blog entry. (no ownership
implied, these are from other YouTube accounts)
The Program POSSUMS
The program
POSSUMS calculates all possible sums created by a set of numbers (2, 3, or
4). With n numbers available, there are
2^n – 1 possible sums.
For n = 2, with
let A and B represent the numbers.
There are 2^2 -1 = 3 possible
sums:
A, B, A + B
For n = 3 (A,
B, C), there are 2^3 – 1 = 7 possible sums:
A, B, C, A + B,
A + C, B + C, A + B + C
For n = 4 (A,
B, C, D) there are 2^4 – 1 = 15 possible sums:
A, B, C, D, A +
B, A + C, A + D, B + C, B + D, B + C, A + B + C, A + B + D, A + C + D, B + C +
D, A + B + C + D
The program
POSSUMS allows all possible sums, including repeats.
TI-84 Plus Program POSSUMS
Menu("SUMS FROM
AVAIL. NUMBERS","2",2,"3",3,"4",4)
Lbl 2
Prompt A,B
{A,B,A+B}→L₁
Goto 5
Lbl 3
Prompt A,B,C
{A,B,C,A+B,A+C,B+C,A+B+C}→L₁
Goto 5
Lbl 4
Prompt A,B,C,D
{A,B,C,D,A+B,A+C,A+D,B+C,B+D,C+D,A+B+C,A+C+D,A+B+D,B+C+D,A+B+C+D}→L₁
Goto 5
Lbl 5
SortA(L₁)
Pause L₁
Examples
3 Numbers: A = 2, B = 3, C = 5
Result: {2, 3, 5, 5, 7, 8, 10}
Notes: There are two 5s, meaning 5 can be made by
two combinations (5, 2 + 3)
4 Numbers: A = 1, B = 6, C = 7, D = 9
Result: {1, 6, 7, 7, 8, 9, 10, 13, 14, 15, 16, 16,
17, 22, 23}
Notes: 7 can be made two ways (7, 6 + 1); 16 can be
made two ways (7 + 9, 1 + 6 + 9)
The Game Show High Rollers
High Rollers
was a game show that aired in three series: 1974-1976, 1978-1980, and
1987-1988. The premise of the game was
to clear as many numbers, ranging from 1 to 9.
In the original 1974-1976 series, each number was attached to a
prize. In the more famous 1978-1980 and
1987-1988 series, the numbers were aligned (seemingly at random) on a 3 x 3
grid. Each column represented a prize or
a group of prizes.
In the main
game there are two contestants. You
would win by either rolling the last number off the board or most likely, force
your opponent to roll a number that can’t be cleared. Obviously the total on the dice is used to
clear numbers. For example, a roll of a
6 (the total counts, not the pips on the individual dies), can clear any of the
following combinations: 6 itself, 1 and 5, 2 and 4, or 1 and 2 and 3. Starting with the 1978 series, rolling
doubles earned the contestant an insurance marker, basically an extra life.
Winning two
games entitled the champion to play the Big Numbers. The object remained the same, get rid of the
numbers 1 to 9 for a major cash prize or car.
One thing to
note: unlike Shut the Box, High Rollers
offered no provision should the last number remaining on the board be a 1.
Below are all
the possible combinations that can be cleared with each roll. There are 61 combinations. Statistically, rolling a 7 is the most likely
event, followed by 6 or 8. However, the
most powerful rolls are 12, followed by 11, then 10.
All the Possible Combos in High Rollers
|
Total
|
Combinations
that can be Cleared
|
|
2
|
2
|
|
3
|
3, 1-2
|
|
4
|
4, 1-3
|
|
5
|
5, 1-4, 2-3
|
|
6
|
6, 1-5, 2-4,
1-2-3
|
|
7
|
7, 1-6, 2-5,
3-4, 1-2-4
|
|
8
|
8, 1-7, 2-6, 3-5,
1-2-5, 1-3-4
|
|
9
|
9, 1-8, 2-7,
3-6, 4-5, 1-2-6, 1-3-5, 2-3-4
|
|
10
|
1-9, 2-8,
3-7, 4-6, 1-2-7, 1-3-6, 1-4-5, 2-3-5, 1-2-3-4
|
|
11
|
2-9, 3-8,
4-7, 5-6, 1-2-8, 1-3-7, 1-4-6, 2-3-6, 2-4-5, 1-2-3-5
|
|
12
|
3-9, 4-8,
5-7, 1-2-9, 1-3-8, 1-4-7, 1-5-6, 2-3-7, 2-4-6, 3-4-5, 1-2-3-6
|
Let’s have some
fun,
Eddie
This blog is
property of Edward Shore, 2017