小测试 | 荒芜知地

前言

发现了游戏里一个有意思的机制：

血量低于80%时，减伤（80%-当前血量%）

无聊就打算建模一下，看看残血减伤有多可怕。

数学分析

假设对方每次伤害都打你 $x\%$ 的血量，且 $x\ge20$ （因为如果打没有20%，绝对是刮痧，不用分析了）

那么：

第一回合

当前血量 $(1-x\%)$

第二回合

当前血量 $(1-x\%-(80\%-(1-x\%))\times x\%) = 1-0.8x\%-(x\%)^2$

即上回合剩的血 - 减伤比例 × 伤害

看起来有点难看，设 $y = x\%$ ，则第二回合当前血量 = $1-0.8y-y^2, (0.2\le y <1 )$
要想活下来，上式结果大于零，解得：

$0.2 \le y < 0.677 = \frac{2}{3}$

也就是说，第二下不死，且对面伤害不太刮痧，第一下打的血在以上区间内。

第 $n$ 回合

归纳可得每回合的血量百分比：

$f(1)=1-y \\ f(2) = f(1)-(0.8-f(1)) \times y \\ ... \\ f(n) = f(n-1) - (0.8-f(n-1)) \times y = (1+y) \times f(n-1) - 0.8y$

好像……不是很好算，那就，跑个代码算吧。

暴力求解

function current_hp(n, y) {
    var cur_hp = [1];
    for (var i = 1; i<=n;i++) {
        if (cur_hp[i-1] >= 0.8) {
        	cur_hp.push(cur_hp[i-1] - y);
    	}
    	else {
        	cur_hp.push((1+y) * cur_hp[i-1] - 0.8 * y);
    	}
    }
    return cur_hp[n];
}
/* 二分法暴力解方程区间 */
function solve_y(n) {
    var l, r, y;
    l = 0.2, r = 1;
    var pre = 0.01;
    while (r - l > pre) {
        y = (l+r)/2;
        if (current_hp(n, y) > 0) {
            l = y;
        }
        else {
            r = y;
        }
    }
    return y;
}
/* 算5个回合的好了 */
for (var i = 1; i < 6; i++) {
    console.log("[round "+i+"] :" + "0.2<=y<"+solve_y(i));
}

结果如下：

[round 1] :0.2<=y<0.99375
[round 2] :0.2<=y<0.6812499999999999
[round 3] :0.2<=y<0.5437500000000001
[round 4] :0.2<=y<0.45625000000000004
[round 5] :0.2<=y<0.40625

所以，如果，对面第一下打你 40%，你可以撑到第五回合且不回血（不考虑伤害浮动和暴击）

更极端一点，比如第 8 回合？

[round 1] :0.2<=y<0.99375
[round 2] :0.2<=y<0.6812499999999999
[round 3] :0.2<=y<0.5437500000000001
[round 4] :0.2<=y<0.45625000000000004
[round 5] :0.2<=y<0.40625
[round 6] :0.2<=y<0.36875
[round 7] :0.2<=y<0.34375
[round 8] :0.2<=y<0.3187500000000001

31% ！生命力非常顽强啊，第一下被打1/3，不回血可以撑 7、8 个回合

假设第一下打 50%，也就是不回血能撑 3 回合，设总血量 22000 ，看看每回合的血量？

for (var i = 1; i < 100000; i++) {
    var res = current_hp(i,0.5);
    if (res >= 0) {
        console.log("[round"+i+"]: "+Math.floor(current_hp(i,0.5)*22000)+"/22000");
    }
    else {
        console.log("[round"+i+"]: "+"0"+"/22000");
        break;
    }
}

结果如下：

[round1]: 11000/22000
[round2]: 7699/22000
[round3]: 2749/22000
[round4]: 0/22000

好像不够明显，看看 40%？

[round1]: 13200/22000
[round2]: 11439/22000
[round3]: 8975/22000
[round4]: 5526/22000
[round5]: 696/22000
[round6]: 0/22000

图像

<html>
<!DOCTYPE html>
<meta charset="utf-8">
<script src="https://d3js.org/d3.v4.js"></script>
<div id="input_group">
<div id = "maxhp">
    <div>血量上限：<a id = "maxhp_text">22000</a></div>
    <input id = "maxhp_slider" type="range" min="12000" max="25000" value="22000" onchange="onChange()">
</div>
<div id = "bhp">
    <div>初始血量：<a id = "bhp_text">22000</a></div>
    <input id = "bhp_slider" type="range" min="1" max="25000" value="22000" onchange="onChange()">
</div>
<div id = "damage">
    <div>对手伤害：<a id = "damage_text">11000</a></div>
    <input id = "damage_slider" type="range" min="1000" max="25000" value="11000" onchange="onChange()">
</div>
<div id = "ro">
    <div>预览回合：<a id = "ro_text">5</a></div>
    <input id = "ro_slider" type="range" min="1" max="15" value="5" onchange="onChange()">
</div>
</div>
<div id="bar_hp"></div>
<script>
var t_r = document.getElementById("ro_text");
var t_m = document.getElementById("maxhp_text");
var t_b = document.getElementById("bhp_text");
var t_d = document.getElementById("damage_text");
var round = +document.getElementById("ro_slider").value;
var max_hp = +document.getElementById("maxhp_slider").value;
var damage0 = +document.getElementById("damage_slider").value;
if (damage0 > max_hp) {
    damage0 = max_hp;
}
var damage = damage0/max_hp;
var begin_hp = +document.getElementById("bhp_slider").value;
t_r.innerText = round;
t_m.innerText = max_hp;
t_b.innerText = begin_hp;
t_d.innerText = damage0;

function onChange(){
    round = +document.getElementById("ro_slider").value;
    max_hp = +document.getElementById("maxhp_slider").value;
    damage0 = +document.getElementById("damage_slider").value;
    if (damage0 > max_hp) {
        damage0 = max_hp;
        document.getElementById("damage_slider").value = max_hp;
    }
    damage = damage0/max_hp;
    begin_hp = +document.getElementById("bhp_slider").value;
    t_r.innerText = round;
    t_m.innerText = max_hp;
    t_b.innerText = begin_hp;
    t_d.innerText = damage0;
    d3.select("svg").remove();
    applyAll();
}

var margin = {top: 20, right: 30, bottom: 40, left: 90},
    width = 460 - margin.left - margin.right,
    height = 400 - margin.top - margin.bottom;

function applyAll() {
    function current_hp(n, y) {
    var cur_hp = [begin_hp/max_hp];
    for (var i = 1; i<=n;i++) {
        if (cur_hp[i-1] >= 0.8) {
        	cur_hp.push(cur_hp[i-1] - y);
    	}
    	else {
        	cur_hp.push((1+y) * cur_hp[i-1] - 0.8 * y);
    	}
    }
    return cur_hp;
}
/* if you want to survive to round n, y max is */
function survive_y(n) {
    var l, r, y;
    l = 0.2, r = 1;
    var pre = 0.01;
    while (r - l > pre) {
        y = (l+r)/2;
        if (current_hp(n, y)[n] > 0) {
            l = y;
        }
        else {
            r = y;
        }
    }
    return y;
}
/* the survive of maximum hp data */
var data1 = [];
for (var i = 1; i <= round; i++){
    var temp = {"round": i, "hp": survive_y(i)};
    data1.push(temp);
}
/* for damage, survive to when? */
var data2 = [];
for (var i = 1; i < 10000000; i++) {
    var res = current_hp(i, damage)[i];
    if (res >= 0) {
        data2.push({"round": i, "hp": res});
    }
    else {
        data2.push({"round": i, "hp": 0});
        break;
    }
}

var svg = d3.select("#bar_hp")
  .append("svg")
    .attr("width", width + margin.left + margin.right)
    .attr("height", height + margin.top + margin.bottom)
  .append("g")
    .attr("transform", "translate(" + margin.left + "," + margin.top + ")");

// Add X axis
  var x = d3.scaleLinear()
    .domain([0, max_hp])
    .range([ 0, width]);
  svg.append("g")
    .attr("transform", "translate(0," + height + ")")
    .call(d3.axisBottom(x))
    .selectAll("text")
      .attr("transform", "translate(-10,0)rotate(-45)")
      .style("text-anchor", "end");

  // Y axis
  var y = d3.scaleBand()
    .range([ 0, height])
    .domain(data1.map(function(d) { return d.round; }))
    .padding(.1);
  svg.append("g")
    .call(d3.axisLeft(y))

  //Bars
  function update() {
    var u = svg.selectAll("myRect")
    .data(data1);
    u.enter()
    .append("rect")
    .attr("x", x(0) )
    .attr("y", function(d) { return y(d.round); })
    .attr("width", function(d) { return x(d.hp)*max_hp; })
    .attr("height", y.bandwidth() )
    .attr("fill", "#69b3a2")
  }
  update();
}

applyAll();

</script>
</html>