An Equation Puzzle

<2020-12-10 Thu>

The other day my kid, Josh, asked me a question: How many equations at most could be correct at the same time on the below toy.

Note: The operators on the second wheel are "+ - * / + - * / + -".

He is only 5-6 years old but has learned much mathematics knowledge. I had just been starting off to teach him programming with Logo and Maxima, so I solved his question with Logo to help his learning. Here is my code:

erase "roll
to roll :start :lst
    output sentence (filter [# >= :start] :lst) (filter [# < :start] :lst)
end

erase "check_eq
to check_eq :eq
    ifelse [and (item 2 :eq) = "/ (item 3 :eq) = 0] [output "false] [output run :eq]
end

erase "check
to check :lsts
    make "ans apply "map fput [(list ?1 ?2 ?3 "= 10 * ?4 + ?5)] :lsts
    make "res map "check_eq :ans
    if [(count filter [?] :res) > 1 ][
        print (sentence [correct equations count:] (count filter [?] :res))
        foreach :ans [show ?]
    ]
end

make "index [1 2 3 4 5 6 7 8 9 10]
make "offsets (crossmap [(list 1 ?1 ?2 ?3 ?4)] :index :index :index :index)

foreach :offsets [
    make "w1 roll (item 1 ?) [0 1 2 3 4 5 6 7 8 9]
    make "w2 roll (item 2 ?) [+ - * / + - * / + -]
    make "w3 roll (item 3 ?) [0 1 2 3 4 5 6 7 8 9]
    make "w4 roll (item 4 ?) [0 1 2 3 4 5 6 7 8 9]
    make "w5 roll (item 5 ?) [0 1 2 3 4 5 6 7 8 9]
    check (list :w1 :w2 :w3 :w4 :w5)
]

The code could be ran using UCB Logo, and it will print the answers which contain more than one correct equation. From the output, I found only one answer who contains 3 (the most) correct equations:

correct equations count: 3
[0 + 9 = 78]
[1 - 0 = 89]
[2 + 1 = 90]
[3 - 2 = 1]
[4 * 3 = 12]
[5 / 4 = 23]
[6 + 5 = 34]
[7 - 6 = 45]
[8 * 7 = 56]
[9 / 8 = 67]

Josh hadn't learned much about Logo yet, maybe only be able to enable the turtle to crawl and draw some geometry shapes. When I showed him the answer and the code, he was excited and asked me to explain the code, but the code is too complex to him so I only taught him what is Cartesian Product (what is function crossmap doing) and he was satisfied.

My plan to teach Josh programming is as follows:

• First, teach him Logo and Maxima
• Then, Lisp(maybe one or many of scheme/common-lisp/elisp/clojure) and Python
• Then, C++, C, and Julia
• After that, let him teach himself

I will write about my teaching and Josh's learning on this site afterward, if you are interested in this sort of contents please stay tuned or contact me via email.