Weekly Rompecabeza: Find the Missing Number

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Dollar Shave Club members are amongst the smartest people in the world. But that doesn’t mean they can’t be smarter. With the help of our friends at American Mensa, DSC is proud to bring you weekly Rompecabezas (Spanish for head breakers) to boost your IQ into the Thermosphere. 

Not gonna lie: This diagram stumped us for a long time when we first saw it. Luckily, we’ve got a couple of crackerjack puzzl-eers (new word?) on staff who were kind enough to figure it out. Now, it’s your turn. Can you decipher the meaning of the icons and determine the missing number?



Need a hint? Here’s a couple:

Hint 1: Each of the icons in the diagram equal a number.

Hint 2: Each row and each column is a mathematical equation, the answer to which can be found at the ends. If you convert each icon into a variable (A, B, and C, for example) can you solve the puzzle?

Stumped? Here’s how we arrived at the answer:



Step 1: Substitute letter variables for the icons

To start, focus on the first row across and first column down. Sub variables for the icons in both equations:

 

1) A + B + C + A = 205

2) A + B + B + B = 205

 

Step 2:  Solve the second equation for A

 

A + B + B + B = 205

A + 3B = 205

A = 205 – 3B

 

Step 3: Go to the first (205) equation & substitute this new value of A (205 – 3B), then solve for C

 

A + B + C + A = 205

(205 – 3B) + B + C + (205 – 3B) = 205

410 – 5B + C = 205

C = 205 – 410 +5B

C = 5B  – 205

 

Step 4: Substitute letter variables for one of the 204 equations

 

A + A + C + C = 204

2A + 2C = 204

 

Step 5: Now substitute your new values for A (205 – 3B) and C (5B – 205) into the equation for 204 (2A + 2C = 204), and solve for the single remaining variable, B

 

2A + 2C = 204

2(205 – 3B) + 2(5B – 205) = 204

410 – 6B + 10B – 410 = 204

4B = 205

B = 51

 

Step 6: Plug this numerical value for B into the A + B + B + B = 205 equation to solve for A

 

A + B + B + B = 205

A + 51 + 51 + 51 = 205

A = 52

 

Step 7: Plug the numerical values for A and B into your original equation to solve for C

 

A + B + C + A = 205

52 + 51 + C + 52 = 205

155 + C = 205

C = 50

 

CONCLUSIONS

 

A = 52

B = 51

C = 50

Now that we know what each icon equals, we can solve for the missing number:
If you didn’t get the answer this week, don’t fret: We’re posting a new rompecabeza each Friday, so check back next week for another chance at puzzle glory!