In Dan Finkel’s TED-Ed video, he shares this math puzzle, paraphrased as follows:
Dr. Schrödinger is creating an army of giant cats for villainous purposes. Your team of secret agents has located his lab, but needs to get through his unusual security system.
The system displays a single number, and has three buttons that control this number:
- Add 5
- Add 7
- Take the square root of the displayed number
Goal: make the numbers 2, 10, and 14 show on the display, in that order.
Rules:
- The display starts at 0.
- It’s fine if other numbers are displayed in between 2, 10, and 14, as long as they appear in that order.
- The system will malfunction if any number is displayed more than once.
- The system will malfunction if any number greater than 60 is displayed.
- The system will malfunction if any fraction/decimal is displayed.
How can you achieve this?
(Sorry, the giant cat army has nothing to do with the puzzle.)
Source: Dan Finkel and TED-Ed
Click to see solution to the giant cat army riddle
You’ll notice that only the square root button makes the number smaller. Since you must make whole numbers, you’ll have to target one of the perfect squares: 4, 9, 16, 25, 36, 49.
You can exclude 25 and 49, because you can already make 5 and 7. You can’t make 4, 9, or 16 with the 5 and 7 buttons. This means you have to aim for 36.
You can do so by hitting: 5, 7, 5, 7, 5, 7. You can use any variation of this except starting with two 5’s or two 7’s, because you need to avoid 10 and 14. The system malfunctions if any number is shown twice, and you need 10 and 14 later.
At 36, you take the square root, landing you at 6. Then you add 5 twice to hit 16. Then you take the square root twice to get to 2. This is the only sequence that gets you here from 36.
To get to 10, you need to get to 3 or 5 first, meaning you need to get to 9 or 25 and take the square root. You can’t reach 25 from 2, so you add 7 to get to 9 instead. Then you take the square root and add 7 to get 10.
To get to 14, you need to get to 7 or 9 first. You already used 9, so you aim for 7 by going for 49 and then taking the square root. You have to make sure not to hit any number you already used. For example, you can add 5 five times and then add 7 twice to get to 49. Then take the square root and add 7. Done!