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Math and Logic

What's in the Circle?

In a circle puzzle like the ones below, dashed arrows mean to add and solid arrows mean to multiply. For example, the solution to the puzzle on the left is a number whose sum is 5+9,5 + 9, which is 14.14. The solution to the puzzle on the right is a number that, when multiplied by 6,6, gives us 12.12. By working backwards, we get a solution of 2.2.

Keep reading to see a few more complicated example puzzles, or dive right into today's challenge if you feel ready.

The two circle puzzles we just saw are both one-step puzzles. To solve multi-step circle puzzles, we might need to use multiple arithmetic steps in the correct order.

The solution number is the sum of 88 and another unknown number. We can find the unknown number since it's the product of 44 and 8,8, which is 32.32. The solution must then be 32+8=40.32 + 8 = \mathbf{40}.

On the other hand, some multi-step puzzles require us to work backwards through multiple steps.

Now, the solution number and 55 add to some unknown number. We need to find that unknown number first. We can do so since it adds to 66 to give us 17.17.

The unknown number was 11,11, and it is the sum of our solution number and 5.5. This means our solution is 6.6.

We can simplify some complex puzzles to make them easier to solve. In the puzzle below, 4848 is the product of three numbers: 2,2, the solution number, and some other unknown number.

Since 2×24=48,2 \times 24 = 48, the solution number and the other unknown number must multiply to 24.24. We can then get rid of the 22 and show 2424 as a product of two numbers.

The puzzle now tells us that 2424 is the product of two numbers: the solution number, and five added to the solution number. Let's list factor pairs of 2424 to find the solution:

24×1(24)×(1)12×2(12)×(2)8×3(8)×(3)6×4(6)×(4) \begin{aligned} 24 & \times 1 &&&&& (-24) \times (-1) \\ 12 & \times 2 &&&&& (-12) \times (-2) \\ 8 & \times 3 &&&&& (-8) \times (-3) \\ 6 & \times 4 &&&&& (-6) \times (-4) \\ \end{aligned}

There are two factor pairs whose numbers have a difference of 5.5. These pairs are (8,3)(8, 3) and (8,3).(-8, -3). This means there are two solutions: 33 and 8.-8.

Can you solve the circle puzzle below?

Today's Challenge

Dashed arrows mean addition, while solid arrows mean multiplication. What number goes in the place of the question mark?

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