This article discusses the math behind the “magical” shoe size/age problem. And it invites you to enter grade schoolers in a “It’s Math Not Magic” competition. The competition coincides with the launch of “Raising the ‘A’ Student”, a LowdCrowd.com initiative that focuses on innovative strategies to promote academic excellence.
A really cool game is going viral on social media. It invites participants to walk through a series of calculations that produce the person’s age and shoe size. The footnote says of the process, “Its shoe – – – – – – – – – magic”.
While slight of hand, optical illusions, and mind twisters are entertaining, they are not “magic”. As for the age/shoe size game, far from it. This goes behind the procedure to provide you the math. If you have not already done so, take a moment to try the problem with your information. It does work, though not by “magic”.
The shoe size/age game is essentially a word problem that translates into a system of Algebraic equations – specifically polynomial equations.
The Output expression is defined in the problem’s last statement, “The first digit(s) are your shoe size & the last 2 digits are your age!” Conversely, the Input expression is embedded in the instructions numbered 1 through 6. Unlocking how this game predicts one’s shoe and age is possible by using the Substitution or Elimination method.
[important]We can confirm the system by plugging various values into the Algebraic expressions. I will extend certain steps to make the illustration more explainable.[/important]
First, we must translate elements of the word problem to Algebraic terms. We will begin by noting variables: Age (Y), Shoe Size (X), and Year Born (Z). As you will later see, we add Current Year (C) in order to make the problem and solution generally applicable.
Second, we must convert any “natural relationship” inherent in the problem. For instance:
(i) As Shoe Size will occupy the thousand and hundred position in the final number, we will need to multiply Age by 100 to arrive at the first term (leading to digits) in the solution.
(ii) Age = Current Year – Year Born. That is, Y = C – Z. And we can redefine this by Z = C – Y and C = Z + Y.
(iii) We can also see a relationship between 1014 and Current Year (C): As C is 2014 at the time of this article, 2014 – 1000 gets us to the 1014. This is important as it allows us to expand the math to other years e.g., next year where C would be 2015. Thus, in simplifying the problem and making it applicable beyond 2014, we replace the 1014 with C – 1000.
Third, we can use the above to construct the Algebraic equation that represents what you should see in the answer; what we are calling Output. That is, given, Output can be written as 100X + Y.
We can refer to Input as the various manipulations performed in the word problem. Namely:
Shoe Size —> X
Multiplied by 5 —> 5X
Add 50 —> 5X + 50
Multiply by 20 —> (5X + 50) * 20
Add 1014 —> ((5x + 50) * 20) + 1014
Subtract the year you were born —> ((5x + 50) * 20) + 1014 – Z.
Hence Input can be expressed as: Input = ((5x + 50) * 20) + 1014 – Z.
We include an additional equation to generalize the solution for all years.
1014 = C – 1000.
Thus, we can state the system as Output and Input equations:
Output: 100X + Y
Input: [(5x + 50) * 20] + 1014 – Z
Generalization: C – 1000
Fourth, we can test the system of Algebraic equations such that Output = Input is another way of stating that Output – Input = 0.
Substitution Method: simplify the model by reducing variables to their relationship with other variables.
Input: 100X + Y = 100X + C – Z
Output: [ ((5X + 50) * 20 + 1014 – Z ] = [ ((5X + 50) * 20 + C – 1000 – Z ]
100X + C – Z = [ 100X + 1000 + C – 1000 – Z ]
100X = [ 100X + 1000 + C – 1000 – Z ] – C + Z
100X = 100X
100X – 100X = 0
Elimination Method: reduce the number of equations by subtraction.
Input = 100X + C – Z
Output = 100X + 1000 + C – 1000 – Z or 100X + C – Z
Input – Output = 100X + C – Z – (100X + C – Z)
Input – Output = 100X + C – Z – 100X – C + Z)
Input – Output = 0 (or Input = Output)
Fifth, let’s replace variables with actual values to test our solution:
[ezcol_1quarter]Shoe Size[/ezcol_1quarter] [ezcol_1quarter]Year Born[/ezcol_1quarter] [ezcol_1quarter]4 Digits[/ezcol_1quarter] [ezcol_1quarter_end]((5X + 50) * 20 + 1014 – Z[/ezcol_1quarter_end][ezcol_1quarter]11[/ezcol_1quarter] [ezcol_1quarter]1950[/ezcol_1quarter] [ezcol_1quarter]1164[/ezcol_1quarter] [ezcol_1quarter_end]1164[/ezcol_1quarter_end] [ezcol_1quarter]12[/ezcol_1quarter] [ezcol_1quarter]1963[/ezcol_1quarter] [ezcol_1quarter]1251[/ezcol_1quarter] [ezcol_1quarter_end]1251[/ezcol_1quarter_end] [ezcol_1quarter]09[/ezcol_1quarter] [ezcol_1quarter]1986[/ezcol_1quarter] [ezcol_1quarter]0928[/ezcol_1quarter] [ezcol_1quarter_end]0928[/ezcol_1quarter_end]
The shoe size/age problem, as stated, has several flaws and limitations. You can test these issues by entering values
1) The first is semantic. The solution is to tell participants his/her age. This is not exactly correct. If, for instance, one is working the problem in November but one’s birthday is in December, the predicted “Age” will be one greater than one’s actual age. Technically, the wording should state, “The first two (2) digits are your shoe size & the last two digits are the age you are or will be after the current year’s birthday.“
2) The next flaw is also a matter of semantics and deals with single-digit shoe size. Namely, the participant must be instructed to include leading zeroes. For instance, a two (2) shoe size technically creates a 3-digit solution with only the first digit for shoe size.
3) Another issue relates to age. As with shoe size, a single-digit age reduces the prediction by one digit. For instance, if this problem were performed in 2014 by someone born in 2006, the age would only occupy one digit. Here again, age must be expressed with leading zeroes. Also, it is plausible that someone older than 99 could perform the problem. A person’s 3-digit age would yield a prediction different than stated in the problem. Hence, without changing the problem, it is limited to persons up to 99 years old.
4) Finally, we noted before that the term “1014” is hard-coded and applicable for he year 2014.Performing this exercise after the year 2014, would yield the incorrect age.
Several enhancements can be made to the problem in order to correct the above. For instance, Input was altered to generalize the solution for other years. That is, rather than using the term 1014, we substituted with the expression “C – 1000″.[/warning]