## The 16 Game

### October 29, 2013

A local store has a promotional game in which scratch-off cards have sixteen spaces that cover a random permutation of the numbers 1 through 16. Customers scratch off the spaces at random. If the scratch-off reveals the number 3, the card loses. If the scratch-offs reveal the numbers 1 and 2, in either order, the card wins. Winning cards receive a discount on store merchandise.

Your task is to write a program that determines the average winning percentage. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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### 4 Responses to “The 16 Game”

1. Jussi Piitulainen said

The only thing that matters is the relative order of 1, 2, and 3 in a permutation. Since 1/3 of the permutations have 1 and 2 before 3, the classical probability of winning is 1/3 and the expected percentage of winning cards in a pack is a little above 33. I didn’t write a program.

2. Solution in Racket.

```#lang racket

(define (shuffle! v)
(define len (vector-length v))
(for ([i (in-range len)])
(define j (+ i (random (- len i))))
(define tmp (vector-ref v i))
(vector-set! v i (vector-ref v j))
(vector-set! v j tmp))
v)

(define slots-per-card 16)

(define (scratch-card card)
(for*/fold ([x 0]) ([i (shuffle (range slots-per-card))]
[c (in-value (vector-ref card i))]
#:break (or (= c 3) (= x 2))
#:when (or (= c 1) (= c 2)))

(define (winning-% [tries 10000])
(define n
(for*/sum ([_ (in-range tries)]
[_ (shuffle! card)]
[x (in-value (scratch-card card))]
#:when (= 2 x))
1))
(exact->inexact (/ n tries)))
```
3. Oops! The winning-% function had a bug in the second for*/sum clause.

```(define (winning-% [tries 10000])
(define n
(for*/sum ([_ (in-range tries)]
[_ (in-value (shuffle! card))] ;; <==
[x (in-value (scratch-card card))]
#:when (= 2 x))
1))
(exact->inexact (/ n tries)))
```
4. Mark said

Trying Nimrod:

```import math

var
numTrials = 1000000
j, tmp: int
a = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
tot = 0

for n in 1..numTrials:
for i in countdown(15,1):
j = random(i+1)
tmp = a[i]
a[i] = a[j]
a[j] = tmp
if a.find(3) > a.find(2) and a.find(3) > a.find(1):
tot += 1

echo(tot / numTrials)
```