Question

Find the missing number.
1 + 2 = 21
2 + 3 = 36
3 + 4 = 43
4 + 5 = ?

Answer 1

Hint: We see that the plus sign in this puzzle does not follow the usual addition. We try to guess a rule using the results of newly defined addition here. We find a rule first and second line. We find another rule for the second and fourth lines, We find use the rule to find the missing number. \[\]

Complete step by step answer:
We are asked to find the missing number in the following puzzle.
1 + 2 = 21
2 + 3 = 36
3 + 4 = 43
4 + 5 = ?
We see that the addition operation here is not our usual addition we know. It is defined in a new way. We see that the operands in the first three addition $ 1,2 $ or $ 2,3 $ or $ 3,4 $ are very small numbers compared to the result of addition 21, 32, or 43 respectively. So multiplication operation is involved in the new rule for addition. \[\]
We try to guess a rule that should work for the first three additions using addition and multiplication. We shall not find any rule that satisfies the first three lines. We now observe alternating additions from top to bottom. We see in the first and third addition that the second operand is multiplied 10 and added to the first operand. Mathematically,
\[\begin{align}
  & 1+2=2\times 10+1=21 \\
 & 3+4=4\times 10+3=43 \\
\end{align}\]
We see in the second addition that we have to follow the same rule as for the first and third addition but we have to add 4 in the result. It means;
\[2+3=\left( 3\times 10+2 \right)+4=32+4=36\]
So we follow the above rule and observe the fourth addition $ 4+5=? $ where we are given operands but not result. So we multiply 10 with second operand 5 add the first operand 4 and then add 4 again with the result to get 58, which means
\[4+5=\left( 5\times 10+4 \right)+4=58\]
So the missing number is 58.\[\]

Note:
 We note that the addition rule for two numbers can be defined as $ a+b=10b+a $ and $ a+b=\left( 10b+a \right)+b $ alternatively. We can also follow a similar rule $ a+b=a\times b+9 $ and $ a+b=a\times b+13 $ alternatively to find the same result. We also see the results of the additions here are in alternating sequence $ 21,36,43,58,65,82,... $