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Verifying calculation rules

I’m ensuring clarity on operations without using unary negation and managing up to 150k rational expressions.

Identifying vertices

I'm trying to identify key points or vertices in the data, using edges to link them, ensuring all rules are followed.

Managing parenthesis structures

I'm working through 91 parenthesis patterns, including canonical forms and normalization. This involves sorting out grouping, syntactic trees, and arithmetic operations like addition and multiplication.

Charting the options

I’m curious about creating a comprehensive list of 20 canonical forms, ensuring all 91 fully parenthesized expressions are included and clearly distinguished.

Validating constraints

I’m thinking through validating parenthesis and ensuring precise concatenation of digits. This leads me to confirm the distinctiveness and count of unique expressions.

Mapping operations

I’m outlining the calculation sequence and grouping operations with parentheses for clarity. Progressing through the sequence, I categorize different parenthesis structures and count the unique configurations for summation and subtraction. The final table and conclusion will include all valid solution families and their respective counts.

Organizing operations

Listing arithmetic expressions and their solutions, creating a table for clarity. Progressing through various operation counts and structures, aiming for a complete look.

Mapping out progress

I’m assured the 20 classes categorize distinct elementary identities. Since common identities standardize values, algebraic methods become unexpectedly intricate.

Assessing intermediate fractions

I’m looking at the intermediate fractions in #11 to ensure they’re negative. The rules exclude fractions and unary minus signs.

Checking constraints

I'm thinking through the impact of intermediate negative values on arithmetic expressions, considering rules and examples involving negative numbers. Progressing towards a clearer understanding.

Piecing together calculations

I’m combining expressions and confirming results with precise fractions and no code. The final table includes category titles and counts, and I’m ensuring all progress is accounted for precisely.

Worked for 2m 43s

20 类解

下表每行给出一类的代表写法，最后一列是该类包含的不同运算结构数。

序号	等式	结构数
1	$(1+2-3+4)\times(5+67)=288$	5
2	$(1+2)\times(34-5+67)=288$	2
3	$(1+23)\times(4+56\div7)=288$	1
4	$(1+23)\times(4-5+6+7)=288$	5
5	$(12-3)\times(45-6-7)=288$	2
6	$(1+2)\times3\times(45-6-7)=288$	4
7	$12\times3\times(4+5+6-7)=288$	10
8	$12\times(3+4\times5-6+7)=288$	5
9	$12\times(3+(4+5-6)\times7)=288$	2
10	$1+2-3+4\times(5+67)=288$	5
11	$1-(2-3-4)\times56+7=288$	4
12	$12+3\times4\times(5\times6-7)=288$	2
13	$12+3+(45-6)\times7=288$	2
14	$12\times3+4\times(56+7)=288$	1
15	$1\times(2+34)\times56\div7=288$	5
16	$(1\times2+34)\times56\div7=288$	2
17	$(1+2)\times3\times4\times56\div7=288$	14
18	$(12-3)\times4\times56\div7=288$	5
19	$12\div3\times4\times(5+6+7)=288$	10
20	$(1+2)\div3\times4\times(5+67)=288$	5
合计	20 类	91

序号

等式

结构数

(1+2-3+4)\times(5+67)=288

(1+2)\times(34-5+67)=288

(1+23)\times(4+56\div7)=288

(1+23)\times(4-5+6+7)=288

(12-3)\times(45-6-7)=288

(1+2)\times3\times(45-6-7)=288

12\times3\times(4+5+6-7)=288

12\times(3+4\times5-6+7)=288

12\times(3+(4+5-6)\times7)=288

1+2-3+4\times(5+67)=288

1-(2-3-4)\times56+7=288

12+3\times4\times(5\times6-7)=288

12+3+(45-6)\times7=288

12\times3+4\times(56+7)=288

1\times(2+34)\times56\div7=288

(1\times2+34)\times56\div7=288

(1+2)\times3\times4\times56\div7=288

(12-3)\times4\times56\div7=288

12\div3\times4\times(5+6+7)=288

(1+2)\div3\times4\times(5+67)=288

合计

20 类

这里怎样算“同一类”？ 只合并连续加减、连续乘除的等价括号写法，例如：

(1+2)\times3\times(45-6-7)

和

(1+2)\times\bigl(3\times(45-(6+7))\bigr)

归为同一类；

a\div(b\div c)

与

a\div b\times c

也按同样原则合并。不会把任意多套一层无效括号算成新解。

全部 91 条逐条展开的等式：
查看完整解答清单

怎样保证没有漏解？

对每一段连续数字，枚举两种情况：直接拼成一个整数；或者在每个可能的位置切开，将左右两段的全部结果分别用加、减、乘、除组合，排除除数为零的情况。递归到单个数字，就覆盖了全部保序的拼数方式、四则运算和括号结构。计算使用精确分数，最终筛出结果恰好为

288

的式子。

注意：上表第 4、11 式的计算涉及负数；若另有“中间不能出现负数”的要求，就需要排除这两类。

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20 类解

怎样保证没有漏解？

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