How old is mommy?
A mother has two children whose ages differ by 5 years. The sum of the squares of their ages is 97. The square of the mother's age can be found by writing the squares of the children's ages one after the other as a four-digit number. How old is the mother?
Can you solve it within 50 seconds?
Hint: Think of the problem as a riddle.
Problem solver solution to the MT calendar problem: Identifying crucial patterns using basic number system concepts and real world experience
The first conclusion easily made from the pattern—sum of squares of ages of two children is less than 100,
Each of the ages must be less than 10.
In other words, the ages are single digit numbers.
This simplifies the problem greatly and it should be possible to identify the exact ages directly without using any formula.
Two patterns now attract attention,
- The sum of squares 97 is very near to but less than 100, the square of 10, and,
- The difference between the two single digit numbers 5 is quite large for the feasible range of values 2 to 9. So contribution of square of the smaller number must be much less than that of the larger one in their sum 97.
The second conclusion made from these two patterns,
The larger age must be as near to 10 as possible.
Making a quick test, you find $8^2+3^2=73$ falling much short of 97.
This confirms the ages of the two children as 9 years and 4 years,
Third conclusion is drawn from the pattern of the square root of first two digits 81 of one possible 4-digit square of mommy's age 8116,
With this combination, mommy's age is surely 90+, practically an impossible figure in real world.
And this finally confirms the square of mommy's age as 1681, square of 41.
Quick verification—square of 40 is 1600. Add 40 and 41 to confirm that,
This is the problem solver solution—all in mind by identifying key patterns and making key conclusions by
- basic idea of squares of single digit and two digit numbers,
- nearness estimation and
- real life experience.
Answer: Mommy's age is 41 years.
This solution takes a maximum of 50 seconds.
It bypasses mathematical deductions altogether.
Key pattern identifications and conclusions based on domain concepts (for this problem, domain is number system) and real world experience lie at the heart of any problem solver solution like this one.
The following solution doesn't bypass maths altogether, but it highlights problem solving concepts and techniques of more variety.
Even though solved in mind, it takes more time.
You may skip it if you choose to do so.
Systematic solution - Stage 1: Create a mathematical model using least number of variables
Assume age of older child to be $x$ and as the child is older by 5 years than the younger one, the age of the younger child is $(x-5)$.
You have taken care of the first statement of the problem.
By the second statement,
If you simplify this relation, you'll land up with a quadratic equation in SINGLE VARIABLE $x$ that CAN HAVE ONLY TWO POSSIBLE VALUES for $x$.
When you examine the nature of the numeric part of the expanded quadratic expression, mentally you can evaluate it to be $-72$ and the coefficients of first and second terms as, $2$ and $-10$.
Still working mentally, the simplified or minimized quadratic equation formed easily as,
Continuing mentally, factorize the LHS expression as $(x-9)(x+4)$ with two roots, $x=9$, valid and $x=-4$ invalid as negative age.
$(x-9)(x+4)=0$ with two roots $x=9$ and $=-4$, an invalid age value.
You have the age of the older child as 9 years (and younger one as 4 years).
Method for quick mental factorization of the quadratic expression
For quick factorization, use your basic concept to form the two factors as $(x-a)(x+b)$ so that numeric term $-ab$ remains negative.
As sum $-a+b=-5$, the coefficient of middle term of the quadratic equation, $a$ must be larger than $b$ by 5, and their product $ab=36$ (negative sign taken care of). What possible values the two may have! Only possibility is 9 and 4 that fits perfectly with the condition.
That's how you get the factorized quadratic equation quickly and in mind as,
$(x-9)(x+4)=0$ with two roots $x=9$ and $=-4$, an invalid age value.
Getting mommy's age: reasoning based on real life knowledge helps to take the right decision
Till now it has been standard math and use of time-saving concept based techniques, and it has been easy enough.
Now when you are required to put squares 81 and 16 of the two ages one after the other, and then take the square root of the four-digit number formed, you identify the pattern instantly that if you put 81 first and then 16 forming 8116, Mommy's age would be 90+.
Ahh, you know that is absolutely abnormal and you would surely go for taking square root of 1681 as 41 (again easy to do in mind), a perfectly normal age for Mommy.
This is the crucial decision you have taken based on real life experience without even trying to take the square root of 8116 (though you have guesstimated the age to be 90+ by examining first two digits 81 in a few seconds—again an intelligent use of time-saving concept based technique).
Answer: Mommy's age is 41.
Main concepts and techniques you have used
Principle of immediate action
This conceptual principle strongly recommends,
Do immediately what you can do now without thinking of any problem that may lie in future.
DO WHAT YOU CAN NOW.
You have ignored the carefully created complexity in the form of Mommy's age and gone right ahead to evaluate the ages of the children.
Math concept on basic nature of roots of a quadratic equation: Using break up of the numeric term first technique
You have used the most basic concepts on relations between the two roots of a quadratic equation,
Sum of the roots equals coefficient of middle term in absolute, and product equals the third numeric term.
Just examine the NUMERIC PRODUCT TERM FIRST trying to break it up into two factors sum of which will equal the coefficient of middle term in absolute and in sign.
This approach is the quickest with least number of trials, whereas breaking up the middle term coefficient into two suitable factors generally would involve many unguided guesses. This is mathematically true as,
Breaking up a number into two suitable additive parts (especially when it is large or negative) is more complex and time-consuming than breaking up a number into two suitable factors—factor combinations possible are always limited in number.
Problem solving in mind
This problem and many other math problems that you are asked to solved can always be solved by writing down steps of the solution following conventional approach.
Instead, if you always try to solve a math problem in mind,
You would think carefully to make no mistake, but you would also use concepts and techniques to cut-through the steps reducing the time to solve as much as possible.
If you make it a habit of solving math problems in mind,
You would find that you are able to find useful patterns and use new techniques to solve problems faster with more confidence.
Focus being on how to solve quicker and with more confidence, your pattern identification and problem solving ability gets stronger.
This habit has a great side-effect—it makes your real life problem solving easier and quicker—because,
After all, most real life problems are to be solved in mind.
Pattern identification technique: Pattern identification from real life experiences
This lies at the heart of quick and effective problem solving,
Identify the key pattern that helps to break through a part of a problem. Use real life experiences coupled with basic subject concepts to identify the key pattern making key conclusions.
First knowing that negative age is invalid, you have chosen 9 for age of older child. And then in a quick scan of first two digits 81 of the four digit square of mommy's age, you could reject the age of 90+ as invalid.
Systematic problem solving
All the above together make up what is Systematic problem solving,
It is in steps using problem solving concepts and techniques that results in quick and confident final solution.