炸鸡块数

麦当劳® 在出售炸鸡块的时候，会分成6、9、20块的包装。据我所知，麦当劳也会在快乐园套餐里面搭配4块炸鸡块，偶尔是10块。本文的目的是描述一个数据问题，而非快消食品的市场实践。

如果你饿了，想吃15块炸鸡，你可以买一个6块装和一个9块装.

如果你极其的饿，你想吃24块炸鸡， 你可以购买两个9块装和一个6块装, 或者你也可以购买4个6块装.

如果你想吃25块炸鸡，那就没办法了，你要么忍着，要么多买点。

我们可以看到有些数字是可以由6、9、20组成的，有些却不行.

可能的炸鸡块数

下面是你要点的炸鸡块数和它对应的可能的组合(使用6、9、20的包装), 按照升序排列：

#Set0 Impossible1 Impossible2 Impossible3 Impossible4 Impossible5 Impossible6 {1,0,0} 7 Impossible8 Impossible9 {0,1,0} 10 Impossible11 Impossible12 {2,0,0} 13 Impossible14 Impossible15 {1,1,0} 16 Impossible17 Impossible18 {0,2,0} {3,0,0} 19 Impossible20 {0,0,1} 21 {2,1,0} 22 Impossible23 Impossible24 {1,2,0} {4,0,0} 25 Impossible26 {1,0,1} 27 {0,3,0} {3,1,0} 28 Impossible29 {0,1,1} 30 {2,2,0} {5,0,0} 31 Impossible32 {2,0,1} 33 {1,3,0} {4,1,0} 34 Impossible35 {1,1,1} 36 {0,4,0} {3,2,0} {6,0,0} 37 Impossible38 {0,2,1} {3,0,1} 39 {2,3,0} {5,1,0} 40 {0,0,2} 41 {2,1,1} 42 {1,4,0} {4,2,0} {7,0,0} 43 Impossible44 {1,2,1} {4,0,1} 45 {0,5,0} {3,3,0} {6,1,0} 46 {1,0,2} 47 {0,3,1} {3,1,1} 48 {2,4,0} {5,2,0} {8,0,0} 49 {0,1,2}  #Set50 {2,2,1} {5,0,1} 51 {1,5,0} {4,3,0} {7,1,0} 52 {2,0,2} 53 {1,3,1} {4,1,1} 54 {0,6,0} {3,4,0} {6,2,0} {9,0,0} 55 {1,1,2} 56 {0,4,1} {3,2,1} {6,0,1} 57 {2,5,0} {5,3,0} {8,1,0} 58 {0,2,2} {3,0,2} 59 {2,3,1} {5,1,1} 60 {0,0,3} {1,6,0} {4,4,0} {7,2,0} {10,0,0} 61 {2,1,2} 62 {1,4,1} {4,2,1} {7,0,1} 63 {0,7,0} {3,5,0} {6,3,0} {9,1,0} 64 {1,2,2} {4,0,2} 65 {0,5,1} {3,3,1} {6,1,1} 66 {1,0,3} {2,6,0} {5,4,0} {8,2,0} {11,0,0} 67 {0,3,2} {3,1,2} 68 {2,4,1} {5,2,1} {8,0,1} 69 {0,1,3} {1,7,0} {4,5,0} {7,3,0} {10,1,0} 70 {2,2,2} {5,0,2} 71 {1,5,1} {4,3,1} {7,1,1} 72 {0,8,0} {2,0,3} {3,6,0} {6,4,0} {9,2,0} {12,0,0}73 {1,3,2} {4,1,2} 74 {0,6,1} {3,4,1} {6,2,1} {9,0,1} 75 {1,1,3} {2,7,0} {5,5,0} {8,3,0} {11,1,0} 76 {0,4,2} {3,2,2} {6,0,2} 77 {2,5,1} {5,3,1} {8,1,1} 78 {0,2,3} {1,8,0} {3,0,3} {4,6,0} {7,4,0} {10,2,0} {13,0,0}79 {2,3,2} {5,1,2} 80 {0,0,4} {1,6,1} {4,4,1} {7,2,1} {10,0,1} 81 {0,9,0} {2,1,3} {3,7,0} {6,5,0} {9,3,0} {12,1,0}82 {1,4,2} {4,2,2} {7,0,2} 83 {0,7,1} {3,5,1} {6,3,1} {9,1,1} 84 {1,2,3} {2,8,0} {4,0,3} {5,6,0} {8,4,0} {11,2,0} {14,0,0}85 {0,5,2} {3,3,2} {6,1,2} 86 {1,0,4} {2,6,1} {5,4,1} {8,2,1} {11,0,1} 87 {0,3,3} {1,9,0} {3,1,3} {4,7,0} {7,5,0} {10,3,0} {13,1,0}88 {2,4,2} {5,2,2} {8,0,2} 89 {0,1,4} {1,7,1} {4,5,1} {7,3,1} {10,1,1} 90 {0,10,0} {2,2,3} {3,8,0} {5,0,3} {6,6,0} {9,4,0} {12,2,0} {15,0,0}91 {1,5,2} {4,3,2} {7,1,2} 92 {0,8,1} {2,0,4} {3,6,1} {6,4,1} {9,2,1} {12,0,1}93 {1,3,3} {2,9,0} {4,1,3} {5,7,0} {8,5,0} {11,3,0} {14,1,0}94 {0,6,2} {3,4,2} {6,2,2} {9,0,2} 95 {1,1,4} {2,7,1} {5,5,1} {8,3,1} {11,1,1} 96 {0,4,3} {1,10,0} {3,2,3} {4,8,0} {6,0,3} {7,6,0} {10,4,0} {13,2,0} {16,0,0}97 {2,5,2} {5,3,2} {8,1,2} 98 {0,2,4} {1,8,1} {3,0,4} {4,6,1} {7,4,1} {10,2,1} {13,0,1}99 {0,11,0} {2,3,3} {3,9,0} {5,1,3} {6,7,0} {9,5,0} {12,3,0} {15,1,0}

43个炸鸡块之后会发生有趣的现象，43之后的数字都可以由6、9、20组成，43是不能由6、9、20组成的最大的数字。通常称这种特殊的数字为Frobenius Number(弗罗贝尼乌斯数) ，也因为很多人使用炸鸡块的例子来说明这种数字，所以它们也被称为Chicken Nugget Number（炸鸡块数）.

对于一个整数集合，只有有限个数字是不能通过这些数字的倍数之和得到，其中最大的数字被称为Frobenius Number(弗罗贝尼乌斯数) 。

先从两个数字的例子讲起

为什么会出现这种现象？为了简化问题，我们假设鸡块只有3和11两种包装。

有了3个鸡块的包装，很容易得到3的倍数的包装(同样，每个11的倍数也容易得到). 并且我们也可以组成11之后每隔3的数字和22之后每隔3的数字 (使用2个11和若干个3即可). 当到达某个点之后，我们可以通过这种方式组成任何数字.

数学家James Sylvester在1884年证明，对于两个质数p, q, 当 n ≥ (p-1)(q-1) 时可以由两个质数组成, 最大的不能组成的数字是 (p-1)(q-1)-1 = pq-p-q

所以, 对于3和11的鸡块包装, Frobenius number是19.

如果多于两个数字，没有确定的公式可以证明. Ferdinand Georg Frobenius (1849-1917)第一次对这种数字进行了研究，因此以他的名字命名。

给麦当劳制造点麻烦?

既然你已经知道了Frobenius，何不在附近的麦当劳点61块炸鸡。

“可以给我来61块炸鸡吗?”

我敢肯定服务员会礼貌的告诉你，他们不卖61快炸鸡，只有20块一盒的包装. 只能卖给你60块。如果你坚持买61块，服务员还是坚持卖60块给你。

几个来回之后你可以理直气壮的说，"其实，你们可以卖给我61块，我点两盒20块的，两盒6块的，和一盒9块的！."

 

*现实中请不要这样，没什么大不了的，这样做不理智!

搞错了？

有时候我看到麦当劳10块装的包装盒，我以为是他们搞错了.

麦当劳真的把11块的包装盒拿来给10块鸡块做广告?(看右边)

然而当我仔细看那个图，我猜发现，左下角的5号和8号实际上是一个鸡块切成了两半。

 

硬币和邮票

类似的现象出现在硬币和邮票的选择上面。应该选择多大面额的硬币?

如果需要将1美元破开，然后在需要美分找零的时候使用，这个问题就不同了。如果我们有1美分，也就是面值$0.01的硬币，那么就可以组成任何面值。则么样才能减少所需要的硬币数 (如果找零给你的$0.99全是1美分的，你肯定受不了!)

即使在美国有$0.5的硬币，却很少用到. 经常被用到的是$0.01, $0.05, $0.10 and $0.25.

下面的表显示在找零时以上4种硬币如何组合。对一每个找零的金额，会有不同的硬币组合方案，下面给出的是最少的硬币组合。

举例. 对于$0.21找零，有9种不同的组合，分别是:21×$0.01 or 1×$0.05,16×$0.01or 2×$0.05, 11×$0.01 or 3×$0.05, 6×$0.01 or 4×$0.05, 1×$0.01 or 1×$0.10, 11×$0.01or 1×$0.10,1×$0.05,6×$0.01 or 1×$0.10, 2×$0.05, 1×$0.01or 2×$0.10,1×$0.01。最优的组合是 2×$0.10, 1×$0.01只用了3枚硬币。

Change# sol</th width=100>MinBest Solution$0.0111 1×$0.01$0.0212 2×$0.01$0.0313 3×$0.01$0.0414 4×$0.01$0.0521 1×$0.05$0.0622 1×$0.05, 1×$0.01$0.0723 1×$0.05, 2×$0.01$0.0824 1×$0.05, 3×$0.01$0.0925 1×$0.05, 4×$0.01$0.1041 1×$0.10$0.1142 1×$0.10, 1×$0.01$0.1243 1×$0.10, 2×$0.01$0.1344 1×$0.10, 3×$0.01$0.1445 1×$0.10, 4×$0.01$0.1562 1×$0.10, 1×$0.05$0.1663 1×$0.10, 1×$0.05, 1×$0.01$0.1764 1×$0.10, 1×$0.05, 2×$0.01$0.1865 1×$0.10, 1×$0.05, 3×$0.01$0.1966 1×$0.10, 1×$0.05, 4×$0.01$0.2092 2×$0.10$0.2193 2×$0.10, 1×$0.01$0.2294 2×$0.10, 2×$0.01$0.2395 2×$0.10, 3×$0.01$0.2496 2×$0.10, 4×$0.01$0.25131 1×$0.25$0.26132 1×$0.25, 1×$0.01$0.27133 1×$0.25, 2×$0.01$0.28134 1×$0.25, 3×$0.01$0.29135 1×$0.25, 4×$0.01$0.30182 1×$0.25, 1×$0.05$0.31183 1×$0.25, 1×$0.05, 1×$0.01$0.32184 1×$0.25, 1×$0.05, 2×$0.01$0.33185 1×$0.25, 1×$0.05, 3×$0.01$0.34186 1×$0.25, 1×$0.05, 4×$0.01$0.35242 1×$0.25, 1×$0.10$0.36243 1×$0.25, 1×$0.10, 1×$0.01$0.37244 1×$0.25, 1×$0.10, 2×$0.01$0.38245 1×$0.25, 1×$0.10, 3×$0.01$0.39246 1×$0.25, 1×$0.10, 4×$0.01$0.40313 1×$0.25, 1×$0.10, 1×$0.05$0.41314 1×$0.25, 1×$0.10, 1×$0.05, 1×$0.01$0.42315 1×$0.25, 1×$0.10, 1×$0.05, 2×$0.01$0.43316 1×$0.25, 1×$0.10, 1×$0.05, 3×$0.01$0.44317 1×$0.25, 1×$0.10, 1×$0.05, 4×$0.01$0.45393 1×$0.25, 2×$0.10$0.46394 1×$0.25, 2×$0.10, 1×$0.01$0.47395 1×$0.25, 2×$0.10, 2×$0.01$0.48396 1×$0.25, 2×$0.10, 3×$0.01$0.49397 1×$0.25, 2×$0.10, 4×$0.01$0.50492 2×$0.25$0.51493 2×$0.25, 1×$0.01$0.52494 2×$0.25, 2×$0.01$0.53495 2×$0.25, 3×$0.01$0.54496 2×$0.25, 4×$0.01$0.55603 2×$0.25, 1×$0.05$0.56604 2×$0.25, 1×$0.05, 1×$0.01$0.57605 2×$0.25, 1×$0.05, 2×$0.01$0.58606 2×$0.25, 1×$0.05, 3×$0.01$0.59607 2×$0.25, 1×$0.05, 4×$0.01$0.60733 2×$0.25, 1×$0.10$0.61734 2×$0.25, 1×$0.10, 1×$0.01$0.62735 2×$0.25, 1×$0.10, 2×$0.01$0.63736 2×$0.25, 1×$0.10, 3×$0.01$0.64737 2×$0.25, 1×$0.10, 4×$0.01$0.65874 2×$0.25, 1×$0.10, 1×$0.05$0.66875 2×$0.25, 1×$0.10, 1×$0.05, 1×$0.01$0.67876 2×$0.25, 1×$0.10, 1×$0.05, 2×$0.01$0.68877 2×$0.25, 1×$0.10, 1×$0.05, 3×$0.01$0.69878 2×$0.25, 1×$0.10, 1×$0.05, 4×$0.01$0.701034 2×$0.25, 2×$0.10$0.711035 2×$0.25, 2×$0.10, 1×$0.01$0.721036 2×$0.25, 2×$0.10, 2×$0.01$0.731037 2×$0.25, 2×$0.10, 3×$0.01$0.741038 2×$0.25, 2×$0.10, 4×$0.01$0.751213 3×$0.25$0.761214 3×$0.25, 1×$0.01$0.771215 3×$0.25, 2×$0.01$0.781216 3×$0.25, 3×$0.01$0.791217 3×$0.25, 4×$0.01$0.801414 3×$0.25, 1×$0.05$0.811415 3×$0.25, 1×$0.05, 1×$0.01$0.821416 3×$0.25, 1×$0.05, 2×$0.01$0.831417 3×$0.25, 1×$0.05, 3×$0.01$0.841418 3×$0.25, 1×$0.05, 4×$0.01$0.851634 3×$0.25, 1×$0.10$0.861635 3×$0.25, 1×$0.10, 1×$0.01$0.871636 3×$0.25, 1×$0.10, 2×$0.01$0.881637 3×$0.25, 1×$0.10, 3×$0.01$0.891638 3×$0.25, 1×$0.10, 4×$0.01$0.901875 3×$0.25, 1×$0.10, 1×$0.05$0.911876 3×$0.25, 1×$0.10, 1×$0.05, 1×$0.01$0.921877 3×$0.25, 1×$0.10, 1×$0.05, 2×$0.01$0.931878 3×$0.25, 1×$0.10, 1×$0.05, 3×$0.01$0.941879 3×$0.25, 1×$0.10, 1×$0.05, 4×$0.01$0.952135 3×$0.25, 2×$0.10$0.962136 3×$0.25, 2×$0.10, 1×$0.01$0.972137 3×$0.25, 2×$0.10, 2×$0.01$0.982138 3×$0.25, 2×$0.10, 3×$0.01$0.992139 3×$0.25, 2×$0.10, 4×$0.01

上面表中最差的情况使用了9枚硬币，在给 $0.94 和 $0.99 找零的时候 (即使我们使用了最优方案).

增加$0.5的硬币

看看下面会有什么样的变化:

Change# sol</th width=100>MinBest Solution$0.0111 1×$0.01$0.0212 2×$0.01$0.0313 3×$0.01$0.0414 4×$0.01$0.0521 1×$0.05$0.0622 1×$0.05, 1×$0.01$0.0723 1×$0.05, 2×$0.01$0.0824 1×$0.05, 3×$0.01$0.0925 1×$0.05, 4×$0.01$0.1041 1×$0.10$0.1142 1×$0.10, 1×$0.01$0.1243 1×$0.10, 2×$0.01$0.1344 1×$0.10, 3×$0.01$0.1445 1×$0.10, 4×$0.01$0.1562 1×$0.10, 1×$0.05$0.1663 1×$0.10, 1×$0.05, 1×$0.01$0.1764 1×$0.10, 1×$0.05, 2×$0.01$0.1865 1×$0.10, 1×$0.05, 3×$0.01$0.1966 1×$0.10, 1×$0.05, 4×$0.01$0.2092 2×$0.10$0.2193 2×$0.10, 1×$0.01$0.2294 2×$0.10, 2×$0.01$0.2395 2×$0.10, 3×$0.01$0.2496 2×$0.10, 4×$0.01$0.25131 1×$0.25$0.26132 1×$0.25, 1×$0.01$0.27133 1×$0.25, 2×$0.01$0.28134 1×$0.25, 3×$0.01$0.29135 1×$0.25, 4×$0.01$0.30182 1×$0.25, 1×$0.05$0.31183 1×$0.25, 1×$0.05, 1×$0.01$0.32184 1×$0.25, 1×$0.05, 2×$0.01$0.33185 1×$0.25, 1×$0.05, 3×$0.01$0.34186 1×$0.25, 1×$0.05, 4×$0.01$0.35242 1×$0.25, 1×$0.10$0.36243 1×$0.25, 1×$0.10, 1×$0.01$0.37244 1×$0.25, 1×$0.10, 2×$0.01$0.38245 1×$0.25, 1×$0.10, 3×$0.01$0.39246 1×$0.25, 1×$0.10, 4×$0.01$0.40313 1×$0.25, 1×$0.10, 1×$0.05$0.41314 1×$0.25, 1×$0.10, 1×$0.05, 1×$0.01$0.42315 1×$0.25, 1×$0.10, 1×$0.05, 2×$0.01$0.43316 1×$0.25, 1×$0.10, 1×$0.05, 3×$0.01$0.44317 1×$0.25, 1×$0.10, 1×$0.05, 4×$0.01$0.45393 1×$0.25, 2×$0.10$0.46394 1×$0.25, 2×$0.10, 1×$0.01$0.47395 1×$0.25, 2×$0.10, 2×$0.01$0.48396 1×$0.25, 2×$0.10, 3×$0.01$0.49397 1×$0.25, 2×$0.10, 4×$0.01$0.50501 1×$0.50$0.51502 1×$0.50, 1×$0.01$0.52503 1×$0.50, 2×$0.01$0.53504 1×$0.50, 3×$0.01$0.54505 1×$0.50, 4×$0.01$0.55622 1×$0.50, 1×$0.05$0.56623 1×$0.50, 1×$0.05, 1×$0.01$0.57624 1×$0.50, 1×$0.05, 2×$0.01$0.58625 1×$0.50, 1×$0.05, 3×$0.01$0.59626 1×$0.50, 1×$0.05, 4×$0.01$0.60772 1×$0.50, 1×$0.10$0.61773 1×$0.50, 1×$0.10, 1×$0.01$0.62774 1×$0.50, 1×$0.10, 2×$0.01$0.63775 1×$0.50, 1×$0.10, 3×$0.01$0.64776 1×$0.50, 1×$0.10, 4×$0.01$0.65933 1×$0.50, 1×$0.10, 1×$0.05$0.66934 1×$0.50, 1×$0.10, 1×$0.05, 1×$0.01$0.67935 1×$0.50, 1×$0.10, 1×$0.05, 2×$0.01$0.68936 1×$0.50, 1×$0.10, 1×$0.05, 3×$0.01$0.69937 1×$0.50, 1×$0.10, 1×$0.05, 4×$0.01$0.701123 1×$0.50, 2×$0.10$0.711124 1×$0.50, 2×$0.10, 1×$0.01$0.721125 1×$0.50, 2×$0.10, 2×$0.01$0.731126 1×$0.50, 2×$0.10, 3×$0.01$0.741127 1×$0.50, 2×$0.10, 4×$0.01$0.751342 1×$0.50, 1×$0.25$0.761343 1×$0.50, 1×$0.25, 1×$0.01$0.771344 1×$0.50, 1×$0.25, 2×$0.01$0.781345 1×$0.50, 1×$0.25, 3×$0.01$0.791346 1×$0.50, 1×$0.25, 4×$0.01$0.801593 1×$0.50, 1×$0.25, 1×$0.05$0.811594 1×$0.50, 1×$0.25, 1×$0.05, 1×$0.01$0.821595 1×$0.50, 1×$0.25, 1×$0.05, 2×$0.01$0.831596 1×$0.50, 1×$0.25, 1×$0.05, 3×$0.01$0.841597 1×$0.50, 1×$0.25, 1×$0.05, 4×$0.01$0.851873 1×$0.50, 1×$0.25, 1×$0.10$0.861874 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.01$0.871875 1×$0.50, 1×$0.25, 1×$0.10, 2×$0.01$0.881876 1×$0.50, 1×$0.25, 1×$0.10, 3×$0.01$0.891877 1×$0.50, 1×$0.25, 1×$0.10, 4×$0.01$0.902184 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.05$0.912185 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.05, 1×$0.01$0.922186 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.05, 2×$0.01$0.932187 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.05, 3×$0.01$0.942188 1×$0.50, 1×$0.25, 1×$0.10, 1×$0.05, 4×$0.01$0.952524 1×$0.50, 1×$0.25, 2×$0.10$0.962525 1×$0.50, 1×$0.25, 2×$0.10, 1×$0.01$0.972526 1×$0.50, 1×$0.25, 2×$0.10, 2×$0.01$0.982527 1×$0.50, 1×$0.25, 2×$0.10, 3×$0.01$0.992528 1×$0.50, 1×$0.25, 2×$0.10, 4×$0.01

增加$0.5的硬币之后，找零 $0.94 和$.99就可以少一枚硬币。

增加$0.50硬币只影响比$0.49大的情况，最优的情况，硬币数只减少了一个 (一个$0.5的替换了两个$0.25).

有更好的方案吗? 如果用不同的面值替换$0.5，会有什么样的结果?

将$0.5的硬币替换成下面的情况...

在这个实验中，保留四个通用面值的硬币: $0.01, $0.05, $0.10 and $0.25 。当使用¢N替换的时候会有什么样的结果变化？

下面是结果表. 左边的N代表我们要增加的硬币，用来替换 $0.50。  'Worst' 这一列代表找零的时候最多使用的硬币数。'Worst Values' 表示使用最多硬币找零的金额。最后， 'Average Change size' 是找零的平均的硬币数.

NWorstWorst ValuesAvg Change Size$0.019 $0.94, $0.99 4.747$0.027 $0.93, $0.94, $0.98, $0.99 3.939$0.037 $0.92, $0.94, $0.97, $0.99 3.939$0.047 $0.92, $0.97, $0.98 3.899$0.059 $0.94, $0.99 4.747$0.067 $0.79, $0.84, $0.89, $0.94, $0.98, $0.99 4.061$0.077 $0.98 3.788$0.087 $0.97 3.758$0.097 $0.92, $0.97 3.828$0.109 $0.94, $0.99 4.747$0.117 $0.89, $0.99 3.879$0.126 $0.78, $0.83, $0.89, $0.91, $0.93, $0.94, $0.96, $0.98 3.697$0.137 $0.92, $0.97 3.838$0.146 $0.82, $0.87, $0.91, $0.93, $0.96, $0.97, $0.98 3.717$0.159 $0.99 4.545$0.166 $0.79, $0.84, $0.88, $0.93, $0.94, $0.97, $0.99 3.768$0.176 $0.83, $0.88, $0.91, $0.96, $0.98, $0.99 3.636$0.186 $0.92, $0.99 3.545$0.196 $0.91, $0.96, $0.97 3.657$0.208 $0.84, $0.89, $0.94, $0.99 4.394$0.216 $0.19, $0.39, $0.59, $0.79, $0.99 3.707$0.226 $0.19 3.525$0.236 $0.19, $0.87 3.535$0.246 $0.19, $0.43, $0.67, $0.91 3.657$0.259 $0.94, $0.99 4.747$0.266 $0.19, $0.24, $0.44, $0.49, $0.69, $0.74, $0.94, $0.99 3.737$0.276 $0.19, $0.24, $0.93, $0.98 3.576$0.286 $0.19, $0.24, $0.97 3.535$0.296 $0.19, $0.24, $0.43, $0.48, $0.67, $0.72, $0.82, $0.91, $0.96 3.747$0.308 $0.99 4.192$0.316 $0.19, $0.24, $0.49, $0.54, $0.79, $0.84 3.626$0.326 $0.19, $0.24 3.495$0.336 $0.19, $0.24 3.525$0.347 $0.92, $0.97 3.869$0.358 $0.94 4.141$0.366 $0.19, $0.24, $0.34, $0.54, $0.59, $0.69, $0.89, $0.94 3.667$0.376 $0.19, $0.24, $0.34, $0.91, $0.96 3.596$0.386 $0.19, $0.24, $0.34, $0.67, $0.72 3.616$0.397 $0.73, $0.97 3.828$0.408 $0.99 4.141$0.416 $0.19, $0.24, $0.34, $0.39, $0.59, $0.64, $0.74, $0.79, $0.99 3.687$0.426 $0.19, $0.24, $0.34, $0.39 3.576$0.436 $0.19, $0.24, $0.34, $0.39, $0.67, $0.72, $0.82 3.657$0.447 $0.68, $0.83 3.869$0.458 $0.89 4.141$0.467 $0.44, $0.89 3.778$0.477 $0.44 3.677$0.487 $0.44, $0.92 3.798$0.498 $0.93 4.141$0.508 $0.94, $0.99 4.242$0.517 $0.44, $0.49, $0.94, $0.99 3.939$0.527 $0.44, $0.49 3.838$0.537 $0.44, $0.49, $0.92, $0.97 3.939$0.548 $0.98 4.081$0.558 $0.99 4.141$0.567 $0.44, $0.49, $0.99 3.899$0.577 $0.44, $0.49 3.838$0.587 $0.44, $0.49, $0.92, $0.97 3.960$0.597 $0.44, $0.49, $0.83, $0.93, $0.98 4.020$0.607 $0.44, $0.49, $0.59, $0.79, $0.84, $0.94, $0.99 4.091$0.617 $0.44, $0.49, $0.59 3.899$0.627 $0.44, $0.49, $0.59 3.869$0.637 $0.44, $0.49, $0.59, $0.97 3.939$0.647 $0.44, $0.49, $0.59, $0.83, $0.88, $0.98 4.051$0.657 $0.44, $0.49, $0.59, $0.64, $0.84, $0.89, $0.99 4.091$0.667 $0.44, $0.49, $0.59, $0.64 3.939$0.677 $0.44, $0.49, $0.59, $0.64 3.929$0.687 $0.44, $0.49, $0.59, $0.64, $0.92 4.000$0.697 $0.44, $0.49, $0.59, $0.64, $0.68, $0.88, $0.93 4.091$0.708 $0.69 4.141$0.718 $0.69 4.020$0.728 $0.69 4.020$0.738 $0.69 4.141$0.748 $0.69 4.182$0.758 $0.69, $0.74 4.242$0.768 $0.69, $0.74 4.141$0.778 $0.69, $0.74 4.141$0.788 $0.69, $0.74 4.182$0.798 $0.69, $0.74 4.202$0.808 $0.69, $0.74 4.242$0.818 $0.69, $0.74 4.182$0.828 $0.69, $0.74 4.202$0.838 $0.69, $0.74 4.212$0.848 $0.69, $0.74 4.242$0.858 $0.69, $0.74, $0.84 4.293$0.868 $0.69, $0.74, $0.84 4.263$0.878 $0.69, $0.74, $0.84 4.273$0.888 $0.69, $0.74, $0.84 4.293$0.898 $0.69, $0.74, $0.84 4.333$0.908 $0.69, $0.74, $0.84, $0.89 4.394$0.918 $0.69, $0.74, $0.84, $0.89 4.384$0.928 $0.69, $0.74, $0.84, $0.89 4.394$0.938 $0.69, $0.74, $0.84, $0.89 4.424$0.948 $0.69, $0.74, $0.84, $0.89, $0.93 4.475$0.959 $0.94 4.545$0.969 $0.94 4.545$0.979 $0.94 4.566$0.989 $0.94 4.606$0.999 $0.94 4.667

结果

我们能够减少最坏情况下的找赎硬币数，如果用指定的硬币替换$0.50，总共有23种选择： $0.12, $0.14, $0.16, $0.17, $0.18, $0.19, $0.21, $0.22, $0.23, $0.24, $0.26, $0.27, $0.28, $0.29, $0.31, $0.32, $0.33, $0.36, $0.37, $0.38, $0.41, $0.42, $0.43

如果我们用以上任何一种面值的硬币替换 $0.50，我们找零的硬币数最多是6个(比用$0.5减少2个).，那要决定用哪个?

取决于你对“最佳”的定义。可以选用 $0.32，它对应的平均找零的硬币数最少。

另外，如果要减少6个硬币的找零时间，应该选择$0.22的硬币.。使用这种硬币，只有当我们要找零$0.19的时候才会需要6枚硬币，其他的找零金额只需要更少的找零金额。

(也可以看到，当N=$0.01, $0.05, $0.10 或者 $0.25, 结果没有变化，也就是说如果增加已有面值的硬币，没有任何意义!)

原文地址：http://datagenetics.com/blog/august22015/index.html