The Magic of Logic (8)
Enders and Hennes, our grandsons, are currently homeschooling. That is caused by the Covid-19 pandemic that has swept around the world. They cannot go to school, cannot meet their best friends, their doting grandparents. I asked Enders, eight years old, whether he was glad he had “holidays”, that he didn’t have to go to school every day. “That’s crazy,” he said, “It is awful!” And brother Hennes, 7, wholeheartedly agreed. They love school.
What they miss, mainly, is the interaction with their friends. The school has not abandoned classes: both boys get daily lessons at home, in the regular school subjects. And sometimes, as a kind of treat, mathematical and logical problems. In these Enders excels, for reasons adequately described in my blog. He has become really, really good at this.
Recently the teacher gave the eight-year-olds a mathematical problem, which I will share with my readers: Can you write equations using only the number four, exactly four times, to get the results from one to ten? Only the basic mathematical operations are allowed: plus, minus, multiply and divide — and brackets. An example made more clear:
Zero can be expressed as 4 : 4 – 4 : 4
or 4 + 4 – 4 – 4
or even (4 – 4) x 4 x 4.
So there are different ways of getting the answer zero using four fours. Now do the same for all numbers from one to ten. That is the task Enders had, and which he solved very quickly.
1 = 4 : 4 – 4 +4
2 = 4 : 4 +4 : 4
3 = (4 x 4 – 4) : 4
4 = (4 – 4) : 4 + 4
5 = (4 x 4 + 4) : 4
6 = (4 + 4) : 4 + 4
7 = 4 + 4 – 4 : 4
8 = 4 + 4 – 4 + 4
9 = 4 + 4 + 4 : 4
10 = (44 – 4) : 4
A compliment to his math teacher (and mother Tanja) that Enders knows perfectly well that multiplication and division take precedence over addition and subtraction (the right “order of operations”). So with 4+4–4:4 you first divide four by four (=1) and then subtract that from 4+4 (=8), which give you 7.
Note that Enders used brackets flawlessly — only where they were absolutely necessary. He also knows that dividing zero by any number gives you zero, e.g. 0:4=0. That makes 4 = (4–4) : 4 + 4 clear to you: (4–4):4=0:4=0. Now add 4 to that and the total is 4.
I’ll confess I struggled myself over the five, and Enders’ solution to ten (“Got it!”) took my breath away. I was nowhere close. Maybe I am not good at this kind of problem. But Enders’ uncle Tommy, the consummate solver, also did not find it easy to compete with the eight-year-old.
Barely relevant addendum: there is a Limerick from my early youth which I must give the two boys:
“There’s a train at 4:4,” said Miss Jenny,
“Four tickets I’ll take — have you any?”
Said the man at the door:
“Not four for four four,
For four for four four is too many!”
“How about with fives?” I asked Enders, and we got to work finding if we could use the number five four times to get all the numbers from one to ten. It is quite hard, and you are never sure if something is impossible, or whether you are simply not finding the right combination. So we consulted Tommy, who did it in his own style: by quickly writing a little program for me that will look for combinations automatically.
Using Tommy’s program we discovered that four fives work for all numbers between 1 and 10 — with the exception of eight! If you use only three fives you can make equations for 2, 4, 5, 6, but not for 1, 3, 7, 8 or 9. If you allow five fives you can get lots of numbers, e.g. 5*5+55/5 = 36. Enders and I will play around with this some more. Four threes work perfectly for 1–10, with some devilishly clever solutions.
Well, the above game is something you can play with young children to wake their interest in elementary mathematics. And yours as well. It is a lot of fun, I can definitely confirm that.
Before I let you go here’s a little problem I gave Enders and Hennes recently:
There are three boxes, one containing only apples, one only oranges and one that has apples and oranges. But some malicious person has switched the labels so that they are now all wrong. You get to reach into any one of the boxes and take out a single fruit. Can you do this and then switch the labels so that they all show correctly what the boxes contain?
Enders solved it very quickly — and enjoyed explaining the solution to everyone in the house. Hennes struggled, but got it in the end. For solving this and the numbers problem above Enders gets two more Gauss Medals. I have printed a number of these on thick and glossy photo paper and cut them out. They are well treasured by the grandsons, and by the students in the local school where I have been doing courses on logic.
I will tell you the solution to the fruit boxes problem (spoiler!): you reach into the A+O box and take out one fruit. If it is an apple it means the box contains only apples (“Apples + Oranges” is a false label) and must be re-labelled “Apples”. Now since the label “Oranges” on the green box is false, it can only contain apples, or apples + oranges. But we have just established that the orange coloured box contains apples, so the green one… Well, finish the solution yourself — re-label the boxes correctly.
In case this is of interest, here are more articles I have written in similar vein:
- The Magic of Logic (1).
- The Magic of Logic (2)
- The Magic of Logic (3)
- The Magic of Logic (4)
- The Magic of Logic (5)
- The Magic of Logic (6)
- The Magic of Logic (7)
- Piet Hein and the Soma Cube
- The Café Wall Illusion
- Tricking the brain
- IQ test: weighing the coins
- Lie to your kids — it’s good for them
- He had a dream — a logical puzzle