April 2019 Menu

Let’s Solve One Together:

Target Practice Plus

“Uncle Bob”

I found this poser type in a daily calendar given to me, I installed it in “the smallest room of the house,” and I found it fun to work on in the mornings. I’ve always liked to play with numbers. I also found that my arithmetic became sharper and more versatile.

I chose here an example medium to hard in difficulty so that it would draw on more strategies for our collaboration. Here it is:

Using {4, 7, 11, 17, 23, 50}, each number at most once, the four basic operations, and grouping to control the sequence, make the target 232.

Strategies

1. The Scope. See what the total of the set is. We find it’s not 232 or even close. This indicates multiplication should be involved. If the set total is much greater than a small target, then division and subtraction should be contemplated.

2. The Toolbox. Take any two of the set and record their sum and difference. In the set above, I find 7 and 23 have a sum of 30 and a difference of 16. Make note of the sums and diffs of all possible pairs. That’s a lot of tools.

3. Products. See if any products come close. For example 4 x 50 gets us 200. Can we go up from there?

4. Check for Factors. Can one of the set divide another? In this case the answer is no, but I see that 4 + 7 divides 11. So if we come within one of the target, without using 4, 7, and 11, we can adjust by +/– one.

5. Parity. We have an even target and mostly odd members – odd primes actually. But two odds sum to an even total.

6. Build Up or Build Down. Sometimes the only way to the target is to overshoot and work down to it. For example, 7 x 50 is 350. Can we pare down from there?

7. Third and Fourth Parties. We have numbers in the toolbox that we created using two of the set. Those tools can be combined for either a solution or for more tools to use. For example (50 + 23) x 4 is 292. It’s closer than 350. The number of possible combinations mounts!

The solutions. My main strategy for this one was #3. The product 23 x 11 is just 21 too high. Subtracting 4 and 17 gives the target 232.

232 = (23 x 11) – 17 – 4.

Looking for a second solution, I did use the example in strategy #7. From 292 I need to drop 60, which can be made from (7 x 11) – 17. This solution employs all six of the set.

232 = (50 + 23) x 4 + 17 – (7 x 11).

Other strategies and other solutions are possible. Please keep me posted. Three targets every month. Better get crackin’.

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