Wednesday, March 15, 2017

High Rollers: Game Show and Possible Sums with the TI-84 Plus



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}→L
Goto 5
Lbl 3
Prompt A,B,C
{A,B,C,A+B,A+C,B+C,A+B+C}→L
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}→L
Goto 5
Lbl 5
SortA(L)
Pause L

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



 



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