在列之间随机分布整数值答案

【问题标题】：Randomly distributing integer values among columns在列之间随机分布整数值
【发布时间】：2012-07-10 11:57:48
【问题描述】：

我正在制作纸牌游戏，我需要预先创建纸牌并将它们存储在 mysql 表中。我试图弄清楚如何制作一个 php 脚本来创建它们。嗯，有 6 个类别的值从 0 到 12 不等，每张卡都有每个类别的值。每张卡片中类别值的总和必须是 36。所以每张卡片是这样的：

6 - 9 - 2 - 10 - 6 - 3

或

0 - 12 - 0 - 12 - 0 - 12

甚至

6 - 6 - 6 - 6 - 6 - 6

我想如果我能随机生成 50 或 60 张卡片，那就太棒了。

提前致谢。

【问题讨论】：

6-6-6-6-6-6? Are you sure that's random?
@MarkByers 你的意思是这样吗？ :P xkcd.com/221
我知道您已经选择了答案，但请查看下面的答案...我认为它比其他解决方案有一些优点。

标签： php mysql math statistics

【解决方案1】：

这样的东西应该可以工作 - 逻辑可能有点不对劲，我还没有测试过。

function generate_card() {
    $max_card = 12;  // What's the highest value of a card number?
    $min_card = 0;   // What's the lowest value of a card number?
    $num_cards = 6;  // How many numbers on a card?
    $total = 36;     // What's the desired sum?
    $numbers = array();
    $running_total = 0;
    for ($i = 0; $i < $num_cards; ++$i) {
        $max = min($max_card, $total - $running_total); //The highest number we can choose
        $min = max($min_card, $total - $running_total - ($max_card * ($num_cards - ($i + 1)))); //The lowest number we can choose

        //- how much is left ? $total - $running total
        //- min is amount_left - ($max_card

        $number = rand($min, $max);
        $running_total += $number;
        $numbers[] = $number;
    }

    return $numbers;
}

编辑：我放错了括号和索引，现在它可以正常工作了。

【讨论】：

假设这个算法有效，它绝对是对我的“天真”算法的改进。我想指出，++$i 是一个非常棘手的结构。为什么不在你实际使用 $i 的地方加 1？
哦，我确实想到了另一件事。我想在这种情况下它是有效的，因为每次迭代的预期值是6，即36/6，但是如果你从0-14 中选择数字，你会得到一种奇怪的偏斜，倾向于在最后使用较低的数字.在这种情况下，这不是问题，但要记住一些事情。在这些情况下，重新排列数字可能会奏效。
@FritsvanCampen - 你所说的“棘手”是什么意思？这只是一个标准的for 循环，尽管我想有些人可能会改用$i++。重新洗牌是确保随机性的好主意。
不。在大多数情况下，它的总和不是 36。
@hbmuller - 它只需要对操作顺序进行一点调整，它现在可以正常工作了。

【解决方案2】：

一个非常幼稚的方法是：

/* return an array of 6 numbers between 0 and 12 that sum to 36 */
generate_card() {
    $numbers = array();
    for($i = 0; $i < 6; $i += 1) {
        $numbers[$i] = rand(0, 12);
    }
    if (array_sum($numbers) !== 36) {
        return generate_card(); // repeat until we find one that does sum to 36
    } else {
        return $numbers; // yay!
    }
}

我认为这对于“简单的纸牌游戏”来说已经足够出色了。

【讨论】：

调整这个随机选择五张牌，然后根据36的多少选择第6张牌。如果1-12的值不可能，然后擦掉所有东西并重新开始。它和以前一样随机，并且节省了很多不必要的递归。
是的，您可以对这个算法进行许多调整。我认为山姆发布了一个更好的虽然。这基本上是你的想法，只是它总是会完成一个循环。
@MarkByers 确实如此，但正如我在山姆回答的 cmets 中解释的那样，在这种情况下这不是问题，因为预期值正好位于中间。
@MarkByers 嗯，是的，虽然我的意思是“随机”和“总和必须是 36”，这才是真正引入偏差的原因。
实际上，要生成大量卡片，我必须反复试验，直到生成卡片为止，但作为一个只需完成一次的过程，没关系。

【解决方案3】：

我知道你已经为此选择了一个答案，但我想我还是会考虑我的解决方案，因为我认为它带来了一些东西。

部分问题在于“随机手”的含义在这里有点模糊。如果你想要一手真正随机的 6 张牌，总点数为 36，你必须有平等的机会从所有总点数为 36 的牌中挑选。

一个数可以相加的方式称为分区。我编写了一个算法来生成 36 个所有可能的分区。那是一万七千多个分区。然后我淘汰了任何使用超过 12 个数字的分区，然后我淘汰了任何涉及超过 6 个数字的分区。这让我有 676 手可能的 6 张牌，总和为 36。

为了确保完全随机性，我将采取的方法是从所有可能的牌中随机挑选。它的存储密集度更高（您必须存储数组），但比建议的任何方法都快得多（因为您只选择一个随机数，然后将其从数组中取出）。这是缩写代码（完整代码如下）：

$allHands = array(
    array(12,12,12,0,0,0),
    array(12,12,11,1,0,0),
    array(12,12,10,2,0,0),
    array(12,12,10,1,1,0),
    // ...etc
);
$hand = $allHands[ rand( 0, count($allHands)-1 ) ];
shuffle( $hand );
// $hand is now a completely random hand of 6 cards summing to 36

这是所有可能牌的完整数组：

$allHands = array(
    array( 12, 12, 12, 0, 0, 0 ),
    array( 12, 12, 11, 1, 0, 0 ),
    array( 12, 12, 10, 2, 0, 0 ),
    array( 12, 12, 10, 1, 1, 0 ),
    array( 12, 12, 9, 3, 0, 0 ),
    array( 12, 12, 9, 2, 1, 0 ),
    array( 12, 12, 9, 1, 1, 1 ),
    array( 12, 12, 8, 4, 0, 0 ),
    array( 12, 12, 8, 3, 1, 0 ),
    array( 12, 12, 8, 2, 2, 0 ),
    array( 12, 12, 8, 2, 1, 1 ),
    array( 12, 12, 7, 5, 0, 0 ),
    array( 12, 12, 7, 4, 1, 0 ),
    array( 12, 12, 7, 3, 2, 0 ),
    array( 12, 12, 7, 3, 1, 1 ),
    array( 12, 12, 7, 2, 2, 1 ),
    array( 12, 12, 6, 6, 0, 0 ),
    array( 12, 12, 6, 5, 1, 0 ),
    array( 12, 12, 6, 4, 2, 0 ),
    array( 12, 12, 6, 4, 1, 1 ),
    array( 12, 12, 6, 3, 3, 0 ),
    array( 12, 12, 6, 3, 2, 1 ),
    array( 12, 12, 6, 2, 2, 2 ),
    array( 12, 12, 5, 5, 2, 0 ),
    array( 12, 12, 5, 5, 1, 1 ),
    array( 12, 12, 5, 4, 3, 0 ),
    array( 12, 12, 5, 4, 2, 1 ),
    array( 12, 12, 5, 3, 3, 1 ),
    array( 12, 12, 5, 3, 2, 2 ),
    array( 12, 12, 4, 4, 4, 0 ),
    array( 12, 12, 4, 4, 3, 1 ),
    array( 12, 12, 4, 4, 2, 2 ),
    array( 12, 12, 4, 3, 3, 2 ),
    array( 12, 12, 3, 3, 3, 3 ),
    array( 12, 11, 11, 2, 0, 0 ),
    array( 12, 11, 11, 1, 1, 0 ),
    array( 12, 11, 10, 3, 0, 0 ),
    array( 12, 11, 10, 2, 1, 0 ),
    array( 12, 11, 10, 1, 1, 1 ),
    array( 12, 11, 9, 4, 0, 0 ),
    array( 12, 11, 9, 3, 1, 0 ),
    array( 12, 11, 9, 2, 2, 0 ),
    array( 12, 11, 9, 2, 1, 1 ),
    array( 12, 11, 8, 5, 0, 0 ),
    array( 12, 11, 8, 4, 1, 0 ),
    array( 12, 11, 8, 3, 2, 0 ),
    array( 12, 11, 8, 3, 1, 1 ),
    array( 12, 11, 8, 2, 2, 1 ),
    array( 12, 11, 7, 6, 0, 0 ),
    array( 12, 11, 7, 5, 1, 0 ),
    array( 12, 11, 7, 4, 2, 0 ),
    array( 12, 11, 7, 4, 1, 1 ),
    array( 12, 11, 7, 3, 3, 0 ),
    array( 12, 11, 7, 3, 2, 1 ),
    array( 12, 11, 7, 2, 2, 2 ),
    array( 12, 11, 6, 6, 1, 0 ),
    array( 12, 11, 6, 5, 2, 0 ),
    array( 12, 11, 6, 5, 1, 1 ),
    array( 12, 11, 6, 4, 3, 0 ),
    array( 12, 11, 6, 4, 2, 1 ),
    array( 12, 11, 6, 3, 3, 1 ),
    array( 12, 11, 6, 3, 2, 2 ),
    array( 12, 11, 5, 5, 3, 0 ),
    array( 12, 11, 5, 5, 2, 1 ),
    array( 12, 11, 5, 4, 4, 0 ),
    array( 12, 11, 5, 4, 3, 1 ),
    array( 12, 11, 5, 4, 2, 2 ),
    array( 12, 11, 5, 3, 3, 2 ),
    array( 12, 11, 4, 4, 4, 1 ),
    array( 12, 11, 4, 4, 3, 2 ),
    array( 12, 11, 4, 3, 3, 3 ),
    array( 12, 10, 10, 4, 0, 0 ),
    array( 12, 10, 10, 3, 1, 0 ),
    array( 12, 10, 10, 2, 2, 0 ),
    array( 12, 10, 10, 2, 1, 1 ),
    array( 12, 10, 9, 5, 0, 0 ),
    array( 12, 10, 9, 4, 1, 0 ),
    array( 12, 10, 9, 3, 2, 0 ),
    array( 12, 10, 9, 3, 1, 1 ),
    array( 12, 10, 9, 2, 2, 1 ),
    array( 12, 10, 8, 6, 0, 0 ),
    array( 12, 10, 8, 5, 1, 0 ),
    array( 12, 10, 8, 4, 2, 0 ),
    array( 12, 10, 8, 4, 1, 1 ),
    array( 12, 10, 8, 3, 3, 0 ),
    array( 12, 10, 8, 3, 2, 1 ),
    array( 12, 10, 8, 2, 2, 2 ),
    array( 12, 10, 7, 7, 0, 0 ),
    array( 12, 10, 7, 6, 1, 0 ),
    array( 12, 10, 7, 5, 2, 0 ),
    array( 12, 10, 7, 5, 1, 1 ),
    array( 12, 10, 7, 4, 3, 0 ),
    array( 12, 10, 7, 4, 2, 1 ),
    array( 12, 10, 7, 3, 3, 1 ),
    array( 12, 10, 7, 3, 2, 2 ),
    array( 12, 10, 6, 6, 2, 0 ),
    array( 12, 10, 6, 6, 1, 1 ),
    array( 12, 10, 6, 5, 3, 0 ),
    array( 12, 10, 6, 5, 2, 1 ),
    array( 12, 10, 6, 4, 4, 0 ),
    array( 12, 10, 6, 4, 3, 1 ),
    array( 12, 10, 6, 4, 2, 2 ),
    array( 12, 10, 6, 3, 3, 2 ),
    array( 12, 10, 5, 5, 4, 0 ),
    array( 12, 10, 5, 5, 3, 1 ),
    array( 12, 10, 5, 5, 2, 2 ),
    array( 12, 10, 5, 4, 4, 1 ),
    array( 12, 10, 5, 4, 3, 2 ),
    array( 12, 10, 5, 3, 3, 3 ),
    array( 12, 10, 4, 4, 4, 2 ),
    array( 12, 10, 4, 4, 3, 3 ),
    array( 12, 9, 9, 6, 0, 0 ),
    array( 12, 9, 9, 5, 1, 0 ),
    array( 12, 9, 9, 4, 2, 0 ),
    array( 12, 9, 9, 4, 1, 1 ),
    array( 12, 9, 9, 3, 3, 0 ),
    array( 12, 9, 9, 3, 2, 1 ),
    array( 12, 9, 9, 2, 2, 2 ),
    array( 12, 9, 8, 7, 0, 0 ),
    array( 12, 9, 8, 6, 1, 0 ),
    array( 12, 9, 8, 5, 2, 0 ),
    array( 12, 9, 8, 5, 1, 1 ),
    array( 12, 9, 8, 4, 3, 0 ),
    array( 12, 9, 8, 4, 2, 1 ),
    array( 12, 9, 8, 3, 3, 1 ),
    array( 12, 9, 8, 3, 2, 2 ),
    array( 12, 9, 7, 7, 1, 0 ),
    array( 12, 9, 7, 6, 2, 0 ),
    array( 12, 9, 7, 6, 1, 1 ),
    array( 12, 9, 7, 5, 3, 0 ),
    array( 12, 9, 7, 5, 2, 1 ),
    array( 12, 9, 7, 4, 4, 0 ),
    array( 12, 9, 7, 4, 3, 1 ),
    array( 12, 9, 7, 4, 2, 2 ),
    array( 12, 9, 7, 3, 3, 2 ),
    array( 12, 9, 6, 6, 3, 0 ),
    array( 12, 9, 6, 6, 2, 1 ),
    array( 12, 9, 6, 5, 4, 0 ),
    array( 12, 9, 6, 5, 3, 1 ),
    array( 12, 9, 6, 5, 2, 2 ),
    array( 12, 9, 6, 4, 4, 1 ),
    array( 12, 9, 6, 4, 3, 2 ),
    array( 12, 9, 6, 3, 3, 3 ),
    array( 12, 9, 5, 5, 5, 0 ),
    array( 12, 9, 5, 5, 4, 1 ),
    array( 12, 9, 5, 5, 3, 2 ),
    array( 12, 9, 5, 4, 4, 2 ),
    array( 12, 9, 5, 4, 3, 3 ),
    array( 12, 9, 4, 4, 4, 3 ),
    array( 12, 8, 8, 8, 0, 0 ),
    array( 12, 8, 8, 7, 1, 0 ),
    array( 12, 8, 8, 6, 2, 0 ),
    array( 12, 8, 8, 6, 1, 1 ),
    array( 12, 8, 8, 5, 3, 0 ),
    array( 12, 8, 8, 5, 2, 1 ),
    array( 12, 8, 8, 4, 4, 0 ),
    array( 12, 8, 8, 4, 3, 1 ),
    array( 12, 8, 8, 4, 2, 2 ),
    array( 12, 8, 8, 3, 3, 2 ),
    array( 12, 8, 7, 7, 2, 0 ),
    array( 12, 8, 7, 7, 1, 1 ),
    array( 12, 8, 7, 6, 3, 0 ),
    array( 12, 8, 7, 6, 2, 1 ),
    array( 12, 8, 7, 5, 4, 0 ),
    array( 12, 8, 7, 5, 3, 1 ),
    array( 12, 8, 7, 5, 2, 2 ),
    array( 12, 8, 7, 4, 4, 1 ),
    array( 12, 8, 7, 4, 3, 2 ),
    array( 12, 8, 7, 3, 3, 3 ),
    array( 12, 8, 6, 6, 4, 0 ),
    array( 12, 8, 6, 6, 3, 1 ),
    array( 12, 8, 6, 6, 2, 2 ),
    array( 12, 8, 6, 5, 5, 0 ),
    array( 12, 8, 6, 5, 4, 1 ),
    array( 12, 8, 6, 5, 3, 2 ),
    array( 12, 8, 6, 4, 4, 2 ),
    array( 12, 8, 6, 4, 3, 3 ),
    array( 12, 8, 5, 5, 5, 1 ),
    array( 12, 8, 5, 5, 4, 2 ),
    array( 12, 8, 5, 5, 3, 3 ),
    array( 12, 8, 5, 4, 4, 3 ),
    array( 12, 8, 4, 4, 4, 4 ),
    array( 12, 7, 7, 7, 3, 0 ),
    array( 12, 7, 7, 7, 2, 1 ),
    array( 12, 7, 7, 6, 4, 0 ),
    array( 12, 7, 7, 6, 3, 1 ),
    array( 12, 7, 7, 6, 2, 2 ),
    array( 12, 7, 7, 5, 5, 0 ),
    array( 12, 7, 7, 5, 4, 1 ),
    array( 12, 7, 7, 5, 3, 2 ),
    array( 12, 7, 7, 4, 4, 2 ),
    array( 12, 7, 7, 4, 3, 3 ),
    array( 12, 7, 6, 6, 5, 0 ),
    array( 12, 7, 6, 6, 4, 1 ),
    array( 12, 7, 6, 6, 3, 2 ),
    array( 12, 7, 6, 5, 5, 1 ),
    array( 12, 7, 6, 5, 4, 2 ),
    array( 12, 7, 6, 5, 3, 3 ),
    array( 12, 7, 6, 4, 4, 3 ),
    array( 12, 7, 5, 5, 5, 2 ),
    array( 12, 7, 5, 5, 4, 3 ),
    array( 12, 7, 5, 4, 4, 4 ),
    array( 12, 6, 6, 6, 6, 0 ),
    array( 12, 6, 6, 6, 5, 1 ),
    array( 12, 6, 6, 6, 4, 2 ),
    array( 12, 6, 6, 6, 3, 3 ),
    array( 12, 6, 6, 5, 5, 2 ),
    array( 12, 6, 6, 5, 4, 3 ),
    array( 12, 6, 6, 4, 4, 4 ),
    array( 12, 6, 5, 5, 5, 3 ),
    array( 12, 6, 5, 5, 4, 4 ),
    array( 12, 5, 5, 5, 5, 4 ),
    array( 11, 11, 11, 3, 0, 0 ),
    array( 11, 11, 11, 2, 1, 0 ),
    array( 11, 11, 11, 1, 1, 1 ),
    array( 11, 11, 10, 4, 0, 0 ),
    array( 11, 11, 10, 3, 1, 0 ),
    array( 11, 11, 10, 2, 2, 0 ),
    array( 11, 11, 10, 2, 1, 1 ),
    array( 11, 11, 9, 5, 0, 0 ),
    array( 11, 11, 9, 4, 1, 0 ),
    array( 11, 11, 9, 3, 2, 0 ),
    array( 11, 11, 9, 3, 1, 1 ),
    array( 11, 11, 9, 2, 2, 1 ),
    array( 11, 11, 8, 6, 0, 0 ),
    array( 11, 11, 8, 5, 1, 0 ),
    array( 11, 11, 8, 4, 2, 0 ),
    array( 11, 11, 8, 4, 1, 1 ),
    array( 11, 11, 8, 3, 3, 0 ),
    array( 11, 11, 8, 3, 2, 1 ),
    array( 11, 11, 8, 2, 2, 2 ),
    array( 11, 11, 7, 7, 0, 0 ),
    array( 11, 11, 7, 6, 1, 0 ),
    array( 11, 11, 7, 5, 2, 0 ),
    array( 11, 11, 7, 5, 1, 1 ),
    array( 11, 11, 7, 4, 3, 0 ),
    array( 11, 11, 7, 4, 2, 1 ),
    array( 11, 11, 7, 3, 3, 1 ),
    array( 11, 11, 7, 3, 2, 2 ),
    array( 11, 11, 6, 6, 2, 0 ),
    array( 11, 11, 6, 6, 1, 1 ),
    array( 11, 11, 6, 5, 3, 0 ),
    array( 11, 11, 6, 5, 2, 1 ),
    array( 11, 11, 6, 4, 4, 0 ),
    array( 11, 11, 6, 4, 3, 1 ),
    array( 11, 11, 6, 4, 2, 2 ),
    array( 11, 11, 6, 3, 3, 2 ),
    array( 11, 11, 5, 5, 4, 0 ),
    array( 11, 11, 5, 5, 3, 1 ),
    array( 11, 11, 5, 5, 2, 2 ),
    array( 11, 11, 5, 4, 4, 1 ),
    array( 11, 11, 5, 4, 3, 2 ),
    array( 11, 11, 5, 3, 3, 3 ),
    array( 11, 11, 4, 4, 4, 2 ),
    array( 11, 11, 4, 4, 3, 3 ),
    array( 11, 10, 10, 5, 0, 0 ),
    array( 11, 10, 10, 4, 1, 0 ),
    array( 11, 10, 10, 3, 2, 0 ),
    array( 11, 10, 10, 3, 1, 1 ),
    array( 11, 10, 10, 2, 2, 1 ),
    array( 11, 10, 9, 6, 0, 0 ),
    array( 11, 10, 9, 5, 1, 0 ),
    array( 11, 10, 9, 4, 2, 0 ),
    array( 11, 10, 9, 4, 1, 1 ),
    array( 11, 10, 9, 3, 3, 0 ),
    array( 11, 10, 9, 3, 2, 1 ),
    array( 11, 10, 9, 2, 2, 2 ),
    array( 11, 10, 8, 7, 0, 0 ),
    array( 11, 10, 8, 6, 1, 0 ),
    array( 11, 10, 8, 5, 2, 0 ),
    array( 11, 10, 8, 5, 1, 1 ),
    array( 11, 10, 8, 4, 3, 0 ),
    array( 11, 10, 8, 4, 2, 1 ),
    array( 11, 10, 8, 3, 3, 1 ),
    array( 11, 10, 8, 3, 2, 2 ),
    array( 11, 10, 7, 7, 1, 0 ),
    array( 11, 10, 7, 6, 2, 0 ),
    array( 11, 10, 7, 6, 1, 1 ),
    array( 11, 10, 7, 5, 3, 0 ),
    array( 11, 10, 7, 5, 2, 1 ),
    array( 11, 10, 7, 4, 4, 0 ),
    array( 11, 10, 7, 4, 3, 1 ),
    array( 11, 10, 7, 4, 2, 2 ),
    array( 11, 10, 7, 3, 3, 2 ),
    array( 11, 10, 6, 6, 3, 0 ),
    array( 11, 10, 6, 6, 2, 1 ),
    array( 11, 10, 6, 5, 4, 0 ),
    array( 11, 10, 6, 5, 3, 1 ),
    array( 11, 10, 6, 5, 2, 2 ),
    array( 11, 10, 6, 4, 4, 1 ),
    array( 11, 10, 6, 4, 3, 2 ),
    array( 11, 10, 6, 3, 3, 3 ),
    array( 11, 10, 5, 5, 5, 0 ),
    array( 11, 10, 5, 5, 4, 1 ),
    array( 11, 10, 5, 5, 3, 2 ),
    array( 11, 10, 5, 4, 4, 2 ),
    array( 11, 10, 5, 4, 3, 3 ),
    array( 11, 10, 4, 4, 4, 3 ),
    array( 11, 9, 9, 7, 0, 0 ),
    array( 11, 9, 9, 6, 1, 0 ),
    array( 11, 9, 9, 5, 2, 0 ),
    array( 11, 9, 9, 5, 1, 1 ),
    array( 11, 9, 9, 4, 3, 0 ),
    array( 11, 9, 9, 4, 2, 1 ),
    array( 11, 9, 9, 3, 3, 1 ),
    array( 11, 9, 9, 3, 2, 2 ),
    array( 11, 9, 8, 8, 0, 0 ),
    array( 11, 9, 8, 7, 1, 0 ),
    array( 11, 9, 8, 6, 2, 0 ),
    array( 11, 9, 8, 6, 1, 1 ),
    array( 11, 9, 8, 5, 3, 0 ),
    array( 11, 9, 8, 5, 2, 1 ),
    array( 11, 9, 8, 4, 4, 0 ),
    array( 11, 9, 8, 4, 3, 1 ),
    array( 11, 9, 8, 4, 2, 2 ),
    array( 11, 9, 8, 3, 3, 2 ),
    array( 11, 9, 7, 7, 2, 0 ),
    array( 11, 9, 7, 7, 1, 1 ),
    array( 11, 9, 7, 6, 3, 0 ),
    array( 11, 9, 7, 6, 2, 1 ),
    array( 11, 9, 7, 5, 4, 0 ),
    array( 11, 9, 7, 5, 3, 1 ),
    array( 11, 9, 7, 5, 2, 2 ),
    array( 11, 9, 7, 4, 4, 1 ),
    array( 11, 9, 7, 4, 3, 2 ),
    array( 11, 9, 7, 3, 3, 3 ),
    array( 11, 9, 6, 6, 4, 0 ),
    array( 11, 9, 6, 6, 3, 1 ),
    array( 11, 9, 6, 6, 2, 2 ),
    array( 11, 9, 6, 5, 5, 0 ),
    array( 11, 9, 6, 5, 4, 1 ),
    array( 11, 9, 6, 5, 3, 2 ),
    array( 11, 9, 6, 4, 4, 2 ),
    array( 11, 9, 6, 4, 3, 3 ),
    array( 11, 9, 5, 5, 5, 1 ),
    array( 11, 9, 5, 5, 4, 2 ),
    array( 11, 9, 5, 5, 3, 3 ),
    array( 11, 9, 5, 4, 4, 3 ),
    array( 11, 9, 4, 4, 4, 4 ),
    array( 11, 8, 8, 8, 1, 0 ),
    array( 11, 8, 8, 7, 2, 0 ),
    array( 11, 8, 8, 7, 1, 1 ),
    array( 11, 8, 8, 6, 3, 0 ),
    array( 11, 8, 8, 6, 2, 1 ),
    array( 11, 8, 8, 5, 4, 0 ),
    array( 11, 8, 8, 5, 3, 1 ),
    array( 11, 8, 8, 5, 2, 2 ),
    array( 11, 8, 8, 4, 4, 1 ),
    array( 11, 8, 8, 4, 3, 2 ),
    array( 11, 8, 8, 3, 3, 3 ),
    array( 11, 8, 7, 7, 3, 0 ),
    array( 11, 8, 7, 7, 2, 1 ),
    array( 11, 8, 7, 6, 4, 0 ),
    array( 11, 8, 7, 6, 3, 1 ),
    array( 11, 8, 7, 6, 2, 2 ),
    array( 11, 8, 7, 5, 5, 0 ),
    array( 11, 8, 7, 5, 4, 1 ),
    array( 11, 8, 7, 5, 3, 2 ),
    array( 11, 8, 7, 4, 4, 2 ),
    array( 11, 8, 7, 4, 3, 3 ),
    array( 11, 8, 6, 6, 5, 0 ),
    array( 11, 8, 6, 6, 4, 1 ),
    array( 11, 8, 6, 6, 3, 2 ),
    array( 11, 8, 6, 5, 5, 1 ),
    array( 11, 8, 6, 5, 4, 2 ),
    array( 11, 8, 6, 5, 3, 3 ),
    array( 11, 8, 6, 4, 4, 3 ),
    array( 11, 8, 5, 5, 5, 2 ),
    array( 11, 8, 5, 5, 4, 3 ),
    array( 11, 8, 5, 4, 4, 4 ),
    array( 11, 7, 7, 7, 4, 0 ),
    array( 11, 7, 7, 7, 3, 1 ),
    array( 11, 7, 7, 7, 2, 2 ),
    array( 11, 7, 7, 6, 5, 0 ),
    array( 11, 7, 7, 6, 4, 1 ),
    array( 11, 7, 7, 6, 3, 2 ),
    array( 11, 7, 7, 5, 5, 1 ),
    array( 11, 7, 7, 5, 4, 2 ),
    array( 11, 7, 7, 5, 3, 3 ),
    array( 11, 7, 7, 4, 4, 3 ),
    array( 11, 7, 6, 6, 6, 0 ),
    array( 11, 7, 6, 6, 5, 1 ),
    array( 11, 7, 6, 6, 4, 2 ),
    array( 11, 7, 6, 6, 3, 3 ),
    array( 11, 7, 6, 5, 5, 2 ),
    array( 11, 7, 6, 5, 4, 3 ),
    array( 11, 7, 6, 4, 4, 4 ),
    array( 11, 7, 5, 5, 5, 3 ),
    array( 11, 7, 5, 5, 4, 4 ),
    array( 11, 6, 6, 6, 6, 1 ),
    array( 11, 6, 6, 6, 5, 2 ),
    array( 11, 6, 6, 6, 4, 3 ),
    array( 11, 6, 6, 5, 5, 3 ),
    array( 11, 6, 6, 5, 4, 4 ),
    array( 11, 6, 5, 5, 5, 4 ),
    array( 11, 5, 5, 5, 5, 5 ),
    array( 10, 10, 10, 6, 0, 0 ),
    array( 10, 10, 10, 5, 1, 0 ),
    array( 10, 10, 10, 4, 2, 0 ),
    array( 10, 10, 10, 4, 1, 1 ),
    array( 10, 10, 10, 3, 3, 0 ),
    array( 10, 10, 10, 3, 2, 1 ),
    array( 10, 10, 10, 2, 2, 2 ),
    array( 10, 10, 9, 7, 0, 0 ),
    array( 10, 10, 9, 6, 1, 0 ),
    array( 10, 10, 9, 5, 2, 0 ),
    array( 10, 10, 9, 5, 1, 1 ),
    array( 10, 10, 9, 4, 3, 0 ),
    array( 10, 10, 9, 4, 2, 1 ),
    array( 10, 10, 9, 3, 3, 1 ),
    array( 10, 10, 9, 3, 2, 2 ),
    array( 10, 10, 8, 8, 0, 0 ),
    array( 10, 10, 8, 7, 1, 0 ),
    array( 10, 10, 8, 6, 2, 0 ),
    array( 10, 10, 8, 6, 1, 1 ),
    array( 10, 10, 8, 5, 3, 0 ),
    array( 10, 10, 8, 5, 2, 1 ),
    array( 10, 10, 8, 4, 4, 0 ),
    array( 10, 10, 8, 4, 3, 1 ),
    array( 10, 10, 8, 4, 2, 2 ),
    array( 10, 10, 8, 3, 3, 2 ),
    array( 10, 10, 7, 7, 2, 0 ),
    array( 10, 10, 7, 7, 1, 1 ),
    array( 10, 10, 7, 6, 3, 0 ),
    array( 10, 10, 7, 6, 2, 1 ),
    array( 10, 10, 7, 5, 4, 0 ),
    array( 10, 10, 7, 5, 3, 1 ),
    array( 10, 10, 7, 5, 2, 2 ),
    array( 10, 10, 7, 4, 4, 1 ),
    array( 10, 10, 7, 4, 3, 2 ),
    array( 10, 10, 7, 3, 3, 3 ),
    array( 10, 10, 6, 6, 4, 0 ),
    array( 10, 10, 6, 6, 3, 1 ),
    array( 10, 10, 6, 6, 2, 2 ),
    array( 10, 10, 6, 5, 5, 0 ),
    array( 10, 10, 6, 5, 4, 1 ),
    array( 10, 10, 6, 5, 3, 2 ),
    array( 10, 10, 6, 4, 4, 2 ),
    array( 10, 10, 6, 4, 3, 3 ),
    array( 10, 10, 5, 5, 5, 1 ),
    array( 10, 10, 5, 5, 4, 2 ),
    array( 10, 10, 5, 5, 3, 3 ),
    array( 10, 10, 5, 4, 4, 3 ),
    array( 10, 10, 4, 4, 4, 4 ),
    array( 10, 9, 9, 8, 0, 0 ),
    array( 10, 9, 9, 7, 1, 0 ),
    array( 10, 9, 9, 6, 2, 0 ),
    array( 10, 9, 9, 6, 1, 1 ),
    array( 10, 9, 9, 5, 3, 0 ),
    array( 10, 9, 9, 5, 2, 1 ),
    array( 10, 9, 9, 4, 4, 0 ),
    array( 10, 9, 9, 4, 3, 1 ),
    array( 10, 9, 9, 4, 2, 2 ),
    array( 10, 9, 9, 3, 3, 2 ),
    array( 10, 9, 8, 8, 1, 0 ),
    array( 10, 9, 8, 7, 2, 0 ),
    array( 10, 9, 8, 7, 1, 1 ),
    array( 10, 9, 8, 6, 3, 0 ),
    array( 10, 9, 8, 6, 2, 1 ),
    array( 10, 9, 8, 5, 4, 0 ),
    array( 10, 9, 8, 5, 3, 1 ),
    array( 10, 9, 8, 5, 2, 2 ),
    array( 10, 9, 8, 4, 4, 1 ),
    array( 10, 9, 8, 4, 3, 2 ),
    array( 10, 9, 8, 3, 3, 3 ),
    array( 10, 9, 7, 7, 3, 0 ),
    array( 10, 9, 7, 7, 2, 1 ),
    array( 10, 9, 7, 6, 4, 0 ),
    array( 10, 9, 7, 6, 3, 1 ),
    array( 10, 9, 7, 6, 2, 2 ),
    array( 10, 9, 7, 5, 5, 0 ),
    array( 10, 9, 7, 5, 4, 1 ),
    array( 10, 9, 7, 5, 3, 2 ),
    array( 10, 9, 7, 4, 4, 2 ),
    array( 10, 9, 7, 4, 3, 3 ),
    array( 10, 9, 6, 6, 5, 0 ),
    array( 10, 9, 6, 6, 4, 1 ),
    array( 10, 9, 6, 6, 3, 2 ),
    array( 10, 9, 6, 5, 5, 1 ),
    array( 10, 9, 6, 5, 4, 2 ),
    array( 10, 9, 6, 5, 3, 3 ),
    array( 10, 9, 6, 4, 4, 3 ),
    array( 10, 9, 5, 5, 5, 2 ),
    array( 10, 9, 5, 5, 4, 3 ),
    array( 10, 9, 5, 4, 4, 4 ),
    array( 10, 8, 8, 8, 2, 0 ),
    array( 10, 8, 8, 8, 1, 1 ),
    array( 10, 8, 8, 7, 3, 0 ),
    array( 10, 8, 8, 7, 2, 1 ),
    array( 10, 8, 8, 6, 4, 0 ),
    array( 10, 8, 8, 6, 3, 1 ),
    array( 10, 8, 8, 6, 2, 2 ),
    array( 10, 8, 8, 5, 5, 0 ),
    array( 10, 8, 8, 5, 4, 1 ),
    array( 10, 8, 8, 5, 3, 2 ),
    array( 10, 8, 8, 4, 4, 2 ),
    array( 10, 8, 8, 4, 3, 3 ),
    array( 10, 8, 7, 7, 4, 0 ),
    array( 10, 8, 7, 7, 3, 1 ),
    array( 10, 8, 7, 7, 2, 2 ),
    array( 10, 8, 7, 6, 5, 0 ),
    array( 10, 8, 7, 6, 4, 1 ),
    array( 10, 8, 7, 6, 3, 2 ),
    array( 10, 8, 7, 5, 5, 1 ),
    array( 10, 8, 7, 5, 4, 2 ),
    array( 10, 8, 7, 5, 3, 3 ),
    array( 10, 8, 7, 4, 4, 3 ),
    array( 10, 8, 6, 6, 6, 0 ),
    array( 10, 8, 6, 6, 5, 1 ),
    array( 10, 8, 6, 6, 4, 2 ),
    array( 10, 8, 6, 6, 3, 3 ),
    array( 10, 8, 6, 5, 5, 2 ),
    array( 10, 8, 6, 5, 4, 3 ),
    array( 10, 8, 6, 4, 4, 4 ),
    array( 10, 8, 5, 5, 5, 3 ),
    array( 10, 8, 5, 5, 4, 4 ),
    array( 10, 7, 7, 7, 5, 0 ),
    array( 10, 7, 7, 7, 4, 1 ),
    array( 10, 7, 7, 7, 3, 2 ),
    array( 10, 7, 7, 6, 6, 0 ),
    array( 10, 7, 7, 6, 5, 1 ),
    array( 10, 7, 7, 6, 4, 2 ),
    array( 10, 7, 7, 6, 3, 3 ),
    array( 10, 7, 7, 5, 5, 2 ),
    array( 10, 7, 7, 5, 4, 3 ),
    array( 10, 7, 7, 4, 4, 4 ),
    array( 10, 7, 6, 6, 6, 1 ),
    array( 10, 7, 6, 6, 5, 2 ),
    array( 10, 7, 6, 6, 4, 3 ),
    array( 10, 7, 6, 5, 5, 3 ),
    array( 10, 7, 6, 5, 4, 4 ),
    array( 10, 7, 5, 5, 5, 4 ),
    array( 10, 6, 6, 6, 6, 2 ),
    array( 10, 6, 6, 6, 5, 3 ),
    array( 10, 6, 6, 6, 4, 4 ),
    array( 10, 6, 6, 5, 5, 4 ),
    array( 10, 6, 5, 5, 5, 5 ),
    array( 9, 9, 9, 9, 0, 0 ),
    array( 9, 9, 9, 8, 1, 0 ),
    array( 9, 9, 9, 7, 2, 0 ),
    array( 9, 9, 9, 7, 1, 1 ),
    array( 9, 9, 9, 6, 3, 0 ),
    array( 9, 9, 9, 6, 2, 1 ),
    array( 9, 9, 9, 5, 4, 0 ),
    array( 9, 9, 9, 5, 3, 1 ),
    array( 9, 9, 9, 5, 2, 2 ),
    array( 9, 9, 9, 4, 4, 1 ),
    array( 9, 9, 9, 4, 3, 2 ),
    array( 9, 9, 9, 3, 3, 3 ),
    array( 9, 9, 8, 8, 2, 0 ),
    array( 9, 9, 8, 8, 1, 1 ),
    array( 9, 9, 8, 7, 3, 0 ),
    array( 9, 9, 8, 7, 2, 1 ),
    array( 9, 9, 8, 6, 4, 0 ),
    array( 9, 9, 8, 6, 3, 1 ),
    array( 9, 9, 8, 6, 2, 2 ),
    array( 9, 9, 8, 5, 5, 0 ),
    array( 9, 9, 8, 5, 4, 1 ),
    array( 9, 9, 8, 5, 3, 2 ),
    array( 9, 9, 8, 4, 4, 2 ),
    array( 9, 9, 8, 4, 3, 3 ),
    array( 9, 9, 7, 7, 4, 0 ),
    array( 9, 9, 7, 7, 3, 1 ),
    array( 9, 9, 7, 7, 2, 2 ),
    array( 9, 9, 7, 6, 5, 0 ),
    array( 9, 9, 7, 6, 4, 1 ),
    array( 9, 9, 7, 6, 3, 2 ),
    array( 9, 9, 7, 5, 5, 1 ),
    array( 9, 9, 7, 5, 4, 2 ),
    array( 9, 9, 7, 5, 3, 3 ),
    array( 9, 9, 7, 4, 4, 3 ),
    array( 9, 9, 6, 6, 6, 0 ),
    array( 9, 9, 6, 6, 5, 1 ),
    array( 9, 9, 6, 6, 4, 2 ),
    array( 9, 9, 6, 6, 3, 3 ),
    array( 9, 9, 6, 5, 5, 2 ),
    array( 9, 9, 6, 5, 4, 3 ),
    array( 9, 9, 6, 4, 4, 4 ),
    array( 9, 9, 5, 5, 5, 3 ),
    array( 9, 9, 5, 5, 4, 4 ),
    array( 9, 8, 8, 8, 3, 0 ),
    array( 9, 8, 8, 8, 2, 1 ),
    array( 9, 8, 8, 7, 4, 0 ),
    array( 9, 8, 8, 7, 3, 1 ),
    array( 9, 8, 8, 7, 2, 2 ),
    array( 9, 8, 8, 6, 5, 0 ),
    array( 9, 8, 8, 6, 4, 1 ),
    array( 9, 8, 8, 6, 3, 2 ),
    array( 9, 8, 8, 5, 5, 1 ),
    array( 9, 8, 8, 5, 4, 2 ),
    array( 9, 8, 8, 5, 3, 3 ),
    array( 9, 8, 8, 4, 4, 3 ),
    array( 9, 8, 7, 7, 5, 0 ),
    array( 9, 8, 7, 7, 4, 1 ),
    array( 9, 8, 7, 7, 3, 2 ),
    array( 9, 8, 7, 6, 6, 0 ),
    array( 9, 8, 7, 6, 5, 1 ),
    array( 9, 8, 7, 6, 4, 2 ),
    array( 9, 8, 7, 6, 3, 3 ),
    array( 9, 8, 7, 5, 5, 2 ),
    array( 9, 8, 7, 5, 4, 3 ),
    array( 9, 8, 7, 4, 4, 4 ),
    array( 9, 8, 6, 6, 6, 1 ),
    array( 9, 8, 6, 6, 5, 2 ),
    array( 9, 8, 6, 6, 4, 3 ),
    array( 9, 8, 6, 5, 5, 3 ),
    array( 9, 8, 6, 5, 4, 4 ),
    array( 9, 8, 5, 5, 5, 4 ),
    array( 9, 7, 7, 7, 6, 0 ),
    array( 9, 7, 7, 7, 5, 1 ),
    array( 9, 7, 7, 7, 4, 2 ),
    array( 9, 7, 7, 7, 3, 3 ),
    array( 9, 7, 7, 6, 6, 1 ),
    array( 9, 7, 7, 6, 5, 2 ),
    array( 9, 7, 7, 6, 4, 3 ),
    array( 9, 7, 7, 5, 5, 3 ),
    array( 9, 7, 7, 5, 4, 4 ),
    array( 9, 7, 6, 6, 6, 2 ),
    array( 9, 7, 6, 6, 5, 3 ),
    array( 9, 7, 6, 6, 4, 4 ),
    array( 9, 7, 6, 5, 5, 4 ),
    array( 9, 7, 5, 5, 5, 5 ),
    array( 9, 6, 6, 6, 6, 3 ),
    array( 9, 6, 6, 6, 5, 4 ),
    array( 9, 6, 6, 5, 5, 5 ),
    array( 8, 8, 8, 8, 4, 0 ),
    array( 8, 8, 8, 8, 3, 1 ),
    array( 8, 8, 8, 8, 2, 2 ),
    array( 8, 8, 8, 7, 5, 0 ),
    array( 8, 8, 8, 7, 4, 1 ),
    array( 8, 8, 8, 7, 3, 2 ),
    array( 8, 8, 8, 6, 6, 0 ),
    array( 8, 8, 8, 6, 5, 1 ),
    array( 8, 8, 8, 6, 4, 2 ),
    array( 8, 8, 8, 6, 3, 3 ),
    array( 8, 8, 8, 5, 5, 2 ),
    array( 8, 8, 8, 5, 4, 3 ),
    array( 8, 8, 8, 4, 4, 4 ),
    array( 8, 8, 7, 7, 6, 0 ),
    array( 8, 8, 7, 7, 5, 1 ),
    array( 8, 8, 7, 7, 4, 2 ),
    array( 8, 8, 7, 7, 3, 3 ),
    array( 8, 8, 7, 6, 6, 1 ),
    array( 8, 8, 7, 6, 5, 2 ),
    array( 8, 8, 7, 6, 4, 3 ),
    array( 8, 8, 7, 5, 5, 3 ),
    array( 8, 8, 7, 5, 4, 4 ),
    array( 8, 8, 6, 6, 6, 2 ),
    array( 8, 8, 6, 6, 5, 3 ),
    array( 8, 8, 6, 6, 4, 4 ),
    array( 8, 8, 6, 5, 5, 4 ),
    array( 8, 8, 5, 5, 5, 5 ),
    array( 8, 7, 7, 7, 7, 0 ),
    array( 8, 7, 7, 7, 6, 1 ),
    array( 8, 7, 7, 7, 5, 2 ),
    array( 8, 7, 7, 7, 4, 3 ),
    array( 8, 7, 7, 6, 6, 2 ),
    array( 8, 7, 7, 6, 5, 3 ),
    array( 8, 7, 7, 6, 4, 4 ),
    array( 8, 7, 7, 5, 5, 4 ),
    array( 8, 7, 6, 6, 6, 3 ),
    array( 8, 7, 6, 6, 5, 4 ),
    array( 8, 7, 6, 5, 5, 5 ),
    array( 8, 6, 6, 6, 6, 4 ),
    array( 8, 6, 6, 6, 5, 5 ),
    array( 7, 7, 7, 7, 7, 1 ),
    array( 7, 7, 7, 7, 6, 2 ),
    array( 7, 7, 7, 7, 5, 3 ),
    array( 7, 7, 7, 7, 4, 4 ),
    array( 7, 7, 7, 6, 6, 3 ),
    array( 7, 7, 7, 6, 5, 4 ),
    array( 7, 7, 7, 5, 5, 5 ),
    array( 7, 7, 6, 6, 6, 4 ),
    array( 7, 7, 6, 6, 5, 5 ),
    array( 7, 6, 6, 6, 6, 5 ),
    array( 6, 6, 6, 6, 6, 6 ),
);

【讨论】：