题面据 Putnam 可核档案整理；中文题意为本站自译，公式请以原始来源为准。 PDF：https://kskedlaya.org/putnam-archive/2016.pdf。
Consider a rectangular region, where and are integers such that . This region is to be tiled using tiles of the two types shown:
$$
\setlength{\unitlength}{4144sp}
\begingroup\makeatletter\ifx\SetFigFont\undefined
\gdef\SetFigFont#1#2#3#4#5{
\reset@font\fontsize{#1}{#2pt}
\fontfamily{#3}\fontseries{#4}\fontshape{#5}
\selectfont}
\fi\endgroup
\begin{picture}(2724,924)(1339,-1423)
\thinlines
{\put(1351,-511){\line( 0,-1){900}}
\put(1351,-1411){\line( 1, 0){450}}
\put(1801,-1411){\line( 0, 1){450}}
\put(1801,-961){\line( 1, 0){450}}
\put(2251,-961){\line( 0, 1){450}}
\put(2251,-511){\line(-1, 0){900}}
}
{\multiput(1351,-961)(128.57143,0.00000){4}{\line( 1, 0){ 64.286}}
\multiput(1801,-961)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
}
{\put(2701,-961){\line( 0,-1){450}}
\put(2701,-1411){\line( 1, 0){900}}
\put(3601,-1411){\line( 0, 1){450}}
\put(3601,-961){\line( 1, 0){450}}
\put(4051,-961){\line( 0, 1){450}}
\put(4051,-511){\line(-1, 0){900}}
\put(3151,-511){\line( 0,-1){450}}
\put(3151,-961){\line(-1, 0){450}}
}
{\multiput(3151,-1411)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
\multiput(3151,-961)(128.57143,0.00000){4}{\line( 1, 0){ 64.286}}
\multiput(3601,-961)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
}
\end{picture}
$$
(The dotted lines divide the tiles into squares.)
The tiles may be rotated and reflected, as long as their sides are parallel to the sides
of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?
考虑一个 矩形区域，其中 和 是整数，使得 。该区域将使用所示的两种类型的图块进行平铺：
$$
\setlength{\unitlength}{4144sp}
\makeatletter\ifx\SetFigFont\undefined
\gdef\SetFigFont#1#2#3#4#5{
\重置@字体\字体大小{#1}{#2pt}
\fontfamily{#3}\fontseries{#4}\fontshape{#5}
\选择字体}
\fi\端基
\开始{图片}(2724,924)(1339,-1423)
\细线
{\put(1351,-511){\line(0,-1){900}}
\put(1351,-1411){\line( 1, 0){450}}
\put(1801,-1411){\line( 0, 1){450}}
\put(1801,-961){\line( 1, 0){450}}
\put(2251,-961){\line( 0, 1){450}}
\put(2251,-511){\line(-1, 0){900}}
}
{\multiput(1351,-961)(128.57143,0.00000){4}{\line( 1, 0){ 64.286}}
\multiput(1801,-961)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
}
{\put(2701,-961){\line(0,-1){450}}
\put(2701,-1411){\line( 1, 0){900}}
\put(3601,-1411){\line( 0, 1){450}}
\put(3601,-961){\line( 1, 0){450}}
\put(4051,-961){\line( 0, 1){450}}
\put(4051,-511){\line(-1, 0){900}}
\put(3151,-511){\line(0,-1){450}}
\put(3151,-961){\line(-1, 0){450}}
}
{\multiput(3151,-1411)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
\multiput(3151,-961)(128.57143,0.00000){4}{\line( 1, 0){ 64.286}}
\multiput(3601,-961)(0.00000,128.57143){4}{\line( 0, 1){ 64.286}}
}
\结束{图片}
$$
(虚线将图块分成 的方块。)
瓷砖可以旋转和反射，只要它们的边与边平行
的矩形区域。它们必须全部适合该区域，并且必须完全覆盖该区域而不重叠。
平铺该区域所需的最小平铺数量是多少？
提示 1
先看同余、整除、最大公因数和 p 进赋值。
提示 2
把整数条件转成同余方程、指数比较或下降过程。
提示 3
若要存在性，用构造；若要唯一性，用最小反例、无限下降或模限制。
完整解答
这页先给题面、题型和提示阶梯，完整证明留给读者逐步展开。2016 年 Putnam A4 可先归入数论：第一步把题设翻成对象、条件、目标三行；第二步沿提示寻找不变量、标准构型或关键变形；第三步补齐边界情形，并回到题目原要求核对。
这题适合先独立想一轮再打开提示。不要急着搜索完整解答，先问自己：题面里最硬的限制是哪一句？