Open Middle®
Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the greatest possible total.
Hint
How would your strategy changed if you started with the answer and then tried to find values you could create to get to it?
Answer
The greatest total we’ve found so far is 83 which comes from (9 x 7) + (5 x 4).
Source: Owen Kaplinsky and Robert Kaplinsky
Traut Core Knowledge 4th Grade Ms. Petros’ class got 96. (9 x 8) + (4 x 6).
That doesn’t work. You used the number 9 twice
I got 115 with ( 9 x 8 ) + ( 7 x 6 )
You can’t have a 3 digit product.
An exhaustive search yields the greatest answer is (9 x 7) + (5 x 4) = 83.
Here is the code:
import itertools
max_answer = 0
valid_answer_count = 0
for trial in itertools.permutations(range(0,10),4):
answer = trial[0]*trial[1]+trial[2]*trial[3]
answer_ones = answer % 10
answer_tens = answer // 10
if (answer_ones in trial): continue
if (answer_tens in trial): continue
if (answer_ones == answer_tens): continue
if (answer > 99): continue
#print(trial,answer)
valid_answer_count += 1
if (answer > max_answer):
max_answer = answer
max_trials = trial
print(“#Valid answers:”, valid_answer_count,”Max answer:”,max_answer,max_trials)
and the result:
#Valid answers: 1144 Max answer: 83 (4, 5, 7, 9)
And here are all the unique answers with a representative expression that results in the answer:
12 = (9*0) + (4*3)
14 = (9*0) + (7*2)
16 = (9*0) + (8*2)
18 = (9*2) + (7*0)
20 = (8*1) + (4*3)
21 = (9*0) + (7*3)
24 = (9*1) + (5*3)
25 = (7*3) + (4*1)
26 = (7*3) + (5*1)
27 = (9*3) + (8*0)
28 = (9*0) + (7*4)
29 = (8*3) + (5*1)
30 = (7*4) + (2*1)
32 = (9*0) + (8*4)
36 = (9*4) + (8*0)
37 = (8*4) + (5*1)
38 = (9*4) + (2*1)
39 = (8*4) + (7*1)
41 = (9*3) + (7*2)
42 = (9*0) + (7*6)
43 = (8*1) + (7*5)
45 = (7*6) + (3*1)
46 = (8*5) + (3*2)
47 = (9*5) + (2*1)
48 = (9*5) + (3*1)
50 = (9*4) + (7*2)
51 = (9*3) + (6*4)
52 = (8*6) + (4*1)
54 = (9*6) + (8*0)
56 = (9*0) + (8*7)
57 = (9*6) + (3*1)
58 = (9*6) + (4*1)
59 = (8*7) + (3*1)
60 = (9*4) + (8*3)
61 = (9*5) + (8*2)
62 = (9*3) + (7*5)
63 = (9*7) + (8*0)
65 = (9*7) + (2*1)
67 = (9*3) + (8*5)
68 = (9*7) + (5*1)
70 = (9*6) + (8*2)
72 = (9*8) + (6*0)
74 = (9*8) + (2*1)
75 = (9*8) + (3*1)
76 = (9*8) + (4*1)
81 = (9*7) + (6*3)
82 = (9*6) + (7*4)
83 = (9*7) + (5*4)
This looks great! I’m assuming you used Python.
My 4th grade students offer this solution:
(9 X 8) + (6 X 4) = 72 + 24 = 96
Good try, but you used the 9 and the 6 both as factors and as digits in the product. You can’t repeat any of the digits used in the factors as digits of the product.
96 8 times 9 and 6 times 1
i meamt 6 times 4 and 8times 9
9 times 7 and 6 times 3 equal 81
(7×3)+(8×1)
(9×8)+(4×5)
(9×8 ) + (5×3) = 87 is what my 4th grader son was able to do.
guess got 8 twice
(9 x 6) + (7 x 5) = 89
54 + 35 = 89
wait didn’t I make 83