July 2026
Problem of the Month

Simply Marble-ous
by Bob Stanton
in GAMES World of Puzzles

For decorating accents in her living room, Lisa bought three bags of marbles, colored red, blue, and green.

Each bag held the same number of marbles.

She emptied the red marbles into the first bowl, the blue marbles into the second bowl, and the green marbles into the third bowl.

To mix them up, she scooped up some (but not all) of the marbles in the first bowl and mixed them into the marbles that were in the second bowl.

Then she scooped up some of the marbles in the second bowl and mixed them into the marbles that were in the third bowl.

Finally, she scooped up some of the marbles in the third bowl and mixed them into the marbles that remained in the first bowl.

She scooped up the same number of marbles each time.

When she was finished, each bowl contained a pretty blend of colors in which no color present in the mixture constituted more than half or less than a fourth of the marbles in the bowl.

When she finished there were 12 red marbles in third bowl.

What was the total number of marbles, how were the colors divided among the bowls, and how many marbles did the scoop hold?

Solution to the Problem:

There were 144 marbles total.

The first and third bowls had 12 red marbles, 12 blue marbles, and 24 green marbles.

The second bowl had 24 red and 24 blue marbles.

The scoop held 36 marbles.

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Here is a breakdown of what happened each time Lisa scooped marbles from one bowl and put them in another:

Initially,
bowl 1: 48 red marbles
bowl 2: 48 blue marbles
bowl 3: 48 green marbles

Lisa scoops 36 red marbles from bowl 1 and mixes them in bowl 2:
bowl 1: 12 red marbles
bowl 2: 48 blue marbles 36 red marbles
bowl 3: 48 green marbles

Lisa scoops 12 red marbles and 24 blue marbles from bowl 2 and mixes them in bowl 3:
bowl 1: 12 red marbles
bowl 2: 24 blue marbles 24 red marbles
bowl 3: 48 green marbles, 12 red marbles, 24 blue marbles

Lisa scoops 12 blue marbles and 24 green marbles from bowl 3 and mixes them in bowl 1:
bowl 1: 12 red marbles, 12 blue marbles, 24 green marbles
bowl 2: 24 blue marbles 24 red marbles
bowl 3: 12 red marbles, 12 blue marbles, 24 green marbles

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Click here for Dr. Kishan's excellent solution

Colin Bowey gets extra credit because he found that the information in the problem does not appear to determine a unique scoop size.   Any whole-number scoop size from 36 to 47 marbles produces the same final distribution.
Here are his results:

Simulation of every possible scoop size
Every scoop size from 36 to 47 works:

S=36
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         36R→
After scoop 1   12R         36R+48B         48G
Scoop 2                     12R+24B→    
After scoop 2   12R         24R+24B     12R+24B+48G
Scoop 3                                 ←0R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G


S=37
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         37R→
After scoop 1   11R         37R+48B         48G
Scoop 2                     13R+24B→    
After scoop 2   11R         24R+24B     13R+24B+48G
Scoop 3                                 ←1R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=38
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         38R→
After scoop 1   10R         38R+48B         48G
Scoop 2                     14R+24B→    
After scoop 2   10R         24R+24B     14R+24B+48G
Scoop 3                                 ←2R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=39
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         39R→
After scoop 1    9R         39R+48B         48G
Scoop 2                     15R+24B→    
After scoop 2    9R         24R+24B     15R+24B+48G
Scoop 3                                 ←3R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=40
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         40R→
After scoop 1    8R         40R+48B         48G
Scoop 2                     16R+24B→    
After scoop 2    8R         24R+24B     16R+24B+48G
Scoop 3                                 ←4R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=41
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         41R→
After scoop 1    7R         41R+48B         48G
Scoop 2                     17R+24B→    
After scoop 2    7R         24R+24B     17R+24B+48G
Scoop 3                                 ←5R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=42
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         42R→
After scoop 1    6R         42R+48B         48G
Scoop 2                     18R+24B→    
After scoop 2    6R         24R+24B     18R+24B+48G
Scoop 3                                 ←6R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=43
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         43R→
After scoop 1    5R         43R+48B         48G
Scoop 2                     19R+24B→    
After scoop 2    5R         24R+24B     19R+24B+48G
Scoop 3                                 ←7R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=44
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         44R→
After scoop 1    4R         44R+48B         48G
Scoop 2                     20R+24B→    
After scoop 2    4R         24R+24B     20R+24B+48G
Scoop 3                                 ←8R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=45
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         45R→
After scoop 1    3R         45R+48B         48G
Scoop 2                     21R+24B→    
After scoop 2    3R         24R+24B     21R+24B+48G
Scoop 3                                 ←9R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=46
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         46R→
After scoop 1    2R          46R+48B        48G
Scoop 2                      22R+24B→    
After scoop 2    2R          24R+24B    22R+24B+48G
Scoop 3                                ←10R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

S=47
Stage          Bowl 1        Bowl 2        Bowl 3
Start           48R           48B           48G
Scoop 1         47R→
After scoop 1    1R         47R+48B         48G
Scoop 2                     23R+24B→    
After scoop 2    1R         24R+24B     23R+24B+48G
Scoop 3                                ←11R+12B+24G        
Final       12R+12B+24G     24R+24B     12R+12B+24G

Correctly solved by:

1. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
2. Seth Cohen	Concord, New Hampshire, USA
3. Colin (Yowie) Bowey	Beechworth, Victoria, Australia

Send any comments or questions to: David Pleacher