Electoral College Math Puzzle
Poll
| No votes (0%) | |||
| No votes (0%) | |||
| No votes (0%) | |||
 | 1 vote (50%) | ||
| No votes (0%) | |||
| | 1 vote (50%) | ||
| No votes (0%) | |||
| | 1 vote (50%) | ||
| | 1 vote (50%) | ||
| | 2 votes (100%) |
2 members have voted
July 27th, 2020 at 11:58:33 AM
permalink
This one is from the Riddler Express.
Riddler Township is having its quadrennial presidential election. Each of the town’s 10 “shires” is allotted a certain number of electoral votes: two, plus one additional vote for every 10 citizens (rounded to the nearest 10).
The names and populations of the 10 shires are summarized in the table below.
As you may know, under this sort of electoral system, it is quite possible for a presidential candidate to lose the popular vote and still win the election.
If there are two candidates running for president of Riddler Township, and every single citizen votes for one or the other, then what is the lowest percentage of the popular vote that a candidate can get while still winning the election?
Riddler Township is having its quadrennial presidential election. Each of the town’s 10 “shires” is allotted a certain number of electoral votes: two, plus one additional vote for every 10 citizens (rounded to the nearest 10).
The names and populations of the 10 shires are summarized in the table below.
| Shire | Population | Electoral Votes |
|---|---|---|
| 1 | 11 | 3 |
| 2 | 21 | 4 |
| 3 | 31 | 5 |
| 4 | 41 | 6 |
| 5 | 51 | 7 |
| 6 | 61 | 8 |
| 7 | 71 | 9 |
| 8 | 81 | 10 |
| 9 | 91 | 11 |
| 10 | 101 | 12 |
| Total | 560 | 75 |
As you may know, under this sort of electoral system, it is quite possible for a presidential candidate to lose the popular vote and still win the election.
If there are two candidates running for president of Riddler Township, and every single citizen votes for one or the other, then what is the lowest percentage of the popular vote that a candidate can get while still winning the election?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 12:27:42 PM
permalink
You need to win 38 electoral votes. To do so you would want to pick the most electoral vote rich shires per person in that shire that add up to 38. That would be shires 1,3,4,5,6,7 with 3,5,6,7,8,9 electoral votes. You would need a minimum of 6,16, 21, 26, 31, and 36 votes to win the respective shires.
And lose all the other shires without a vote. So 136/560, or approximately 24.29%.
(you could also use other shires that add up to 38 electoral votes)
And lose all the other shires without a vote. So 136/560, or approximately 24.29%.
(you could also use other shires that add up to 38 electoral votes)
July 27th, 2020 at 1:18:52 PM
permalink
Quote: SOOPOOYou need to win 38 electoral votes. To do so you would want to pick the most electoral vote rich shires per person in that shire that add up to 38. That would be shires 1,3,4,5,6,7 with 3,5,6,7,8,9 electoral votes. You would need a minimum of 6,16, 21, 26, 31, and 36 votes to win the respective shires.
And lose all the other shires without a vote. So 136/560, or approximately 24.29%.
(you could also use other shires that add up to 38 electoral votes)
I can't do any better than 136 either. My way did it with shires 1, 2, 3, 4, and 6.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 1:30:26 PM
permalink
Quote: WizardQuote: SOOPOOYou need to win 38 electoral votes. To do so you would want to pick the most electoral vote rich shires per person in that shire that add up to 38. That would be shires 1,3,4,5,6,7 with 3,5,6,7,8,9 electoral votes. You would need a minimum of 6,16, 21, 26, 31, and 36 votes to win the respective shires.
And lose all the other shires without a vote. So 136/560, or approximately 24.29%.
(you could also use other shires that add up to 38 electoral votes)I can't do any better than 136 either. My way did it with shires 1, 2, 3, 4, and 6.
1,2,3,4, and 6 would result in a loss. That's only 26 electoral votes.
July 27th, 2020 at 1:38:19 PM
permalink
Quote: SOOPOO1,2,3,4, and 6 would result in a loss. That's only 26 electoral votes.
Oops, I forgot to include Shire 10.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 1:39:35 PM
permalink
I think a variety of similar ideas work,...Quote: SOOPOO...1,2,3,4, and 6 would result in a loss. That's only 26 electoral votes.
...you just have to use nearly all the smaller ones and one of the biggest four. {3+4+5+6+7+8}+{9}=42, so using "9" you can ignore the "4". If you use "10 "ignore the "5" etc.
July 27th, 2020 at 2:28:47 PM
permalink
24.29%
July 27th, 2020 at 2:31:34 PM
permalink
Yay, I think this is one of the few puzzles I am confident I got right here. Lol
July 27th, 2020 at 3:25:21 PM
permalink
The ratio of electoral vote to population increases monotonically from Shire 1 to 10.
So we want the lowest set of shires that add up to 38 elcetoral votes.
Shires 1, 2, 3, 4 add up to 18 electoral votes. We need 20 more. I'll take 6 (8 votes) and 10 (12 electoral votes.)
Shires # 1,2,3,4 6 and 10 have 38/75 electoral votes and 265/560 popular votes: which is 47.3214% of the popular vote.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 27th, 2020 at 3:35:06 PM
permalink
Quote: gordonm888
The ratio of electoral vote to population increases monotonically from Shire 1 to 10.
So we want the lowest set of shires that add up to 38 elcetoral votes.
Shires 1, 2, 3, 4 add up to 18 electoral votes. We need 20 more. I'll take 6 (8 votes) and 10 (12 electoral votes.)
Shires # 1,2,3,4 6 and 10 have 38/75 electoral votes and 265/560 popular votes: which is 47.3214% of the popular vote.
Gordon, I think you're assuming everyone in any given shire votes the same way. This is not necessarily the case.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 3:36:27 PM
permalink
Same problem but the 2020 U.S. election. Here is a table to help you. Populations as of July 2019. Remember, electoral votes are reset only every ten years.
As a reminder, there is a forum rule against political statements.
As a reminder, there is a forum rule against political statements.
| State | Population | Electoral votes |
|---|---|---|
| California | 39,512,223 | 55 |
| Texas | 28,995,881 | 38 |
| Florida | 21,477,737 | 29 |
| New York | 19,453,561 | 29 |
| Illinois | 12,671,821 | 20 |
| Pennsylvania | 12,801,989 | 20 |
| Ohio | 11,689,100 | 18 |
| Georgia | 10,617,423 | 16 |
| North Carolina | 10,488,084 | 15 |
| Michigan | 9,986,857 | 16 |
| New Jersey | 8,882,190 | 14 |
| Virginia | 8,535,519 | 13 |
| Washington | 7,614,893 | 12 |
| Arizona | 7,278,717 | 11 |
| Massachusetts | 6,949,503 | 11 |
| Tennessee | 6,833,174 | 11 |
| Indiana | 6,732,219 | 11 |
| Missouri | 6,137,428 | 10 |
| Maryland | 6,045,680 | 10 |
| Wisconsin | 5,822,434 | 10 |
| Colorado | 5,758,736 | 9 |
| Minnesota | 5,639,632 | 10 |
| South Carolina | 5,148,714 | 9 |
| Alabama | 4,903,185 | 9 |
| Louisiana | 4,648,794 | 8 |
| Kentucky | 4,467,673 | 8 |
| Oregon | 4,217,737 | 7 |
| Oklahoma | 3,956,971 | 7 |
| Connecticut | 3,565,287 | 7 |
| Utah | 3,205,958 | 6 |
| Iowa | 3,155,070 | 6 |
| Nevada | 3,080,156 | 6 |
| Arkansas | 3,017,825 | 6 |
| Mississippi | 2,976,149 | 6 |
| Kansas | 2,913,314 | 6 |
| New Mexico | 2,096,829 | 5 |
| Nebraska | 1,934,408 | 5 |
| West Virginia | 1,792,147 | 5 |
| Idaho | 1,787,065 | 4 |
| Hawaii | 1,415,872 | 4 |
| New Hampshire | 1,359,711 | 4 |
| Maine | 1,344,212 | 4 |
| Montana | 1,068,778 | 3 |
| Rhode Island | 1,059,361 | 4 |
| Delaware | 973,764 | 3 |
| South Dakota | 884,659 | 3 |
| North Dakota | 762,062 | 3 |
| Alaska | 731,545 | 3 |
| DC | 705,749 | 3 |
| Vermont | 623,989 | 3 |
| Wyoming | 578,759 | 3 |
| Total | 328,300,544 | 538 |
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 3:37:59 PM
permalink
The answer is 136 / 560 = 17 / 70; there are six ways to get it
1, 3, 4, 5, 6, 7
1, 2, 4, 5, 6, 8
1, 2, 3, 5, 7, 8
1, 2, 3, 5, 6, 9
1, 2, 3, 4, 7, 9
1, 2, 3, 4, 6, 10
Note that Shire N has 10N + 1 voters and N + 2 electoral votes; you need 5N + 1 voters to get the N + 2 electoral votes from Shire N.
Let S1, S2, S3, S4, S5, S6 be the numbers of six Shires with 38 electoral votes
The total number of votes needed = (5 S1 + 1) + (5 S2 + 1) + (5 S3 + 1) + (5 S4 + 1) + (5 S5 + 1) + (5 S6 + 1) = 5 (S1 + S2 + S3 + S4 + S5 + S6) + 6, which equals 136 when S1 + S2 + S3 + S4 + S5 + S6 = 26.
The number of electoral votes = (S1 + 2) + (S2 + 2) + (S3 + 2) + (S4 + 2) + (S5 + 2) + (S6 + 2) = S1 + S2 + S3 + S4 + S5 + S6 + 12 = 38, so S1 + S2 + S3 + S4 + S5 + S6 = 26.
Thus, any combination of six Shires whose numbers add up to 26 works.
July 27th, 2020 at 4:03:59 PM
permalink
Quote: WizardSame problem but the 2020 U.S. election. Here is a table to help you. Populations as of July 2019. Remember, electoral votes are reset only every ten years.
As a reminder, there is a forum rule against political statements.
State Population Electoral votes California 39,512,223 55 Texas 28,995,881 38 Florida 21,477,737 29 New York 19,453,561 29 Illinois 12,671,821 20 Pennsylvania 12,801,989 20 Ohio 11,689,100 18 Georgia 10,617,423 16 North Carolina 10,488,084 15 Michigan 9,986,857 16 New Jersey 8,882,190 14 Virginia 8,535,519 13 Washington 7,614,893 12 Arizona 7,278,717 11 Massachusetts 6,949,503 11 Tennessee 6,833,174 11 Indiana 6,732,219 11 Missouri 6,137,428 10 Maryland 6,045,680 10 Wisconsin 5,822,434 10 Colorado 5,758,736 9 Minnesota 5,639,632 10 South Carolina 5,148,714 9 Alabama 4,903,185 9 Louisiana 4,648,794 8 Kentucky 4,467,673 8 Oregon 4,217,737 7 Oklahoma 3,956,971 7 Connecticut 3,565,287 7 Utah 3,205,958 6 Iowa 3,155,070 6 Nevada 3,080,156 6 Arkansas 3,017,825 6 Mississippi 2,976,149 6 Kansas 2,913,314 6 New Mexico 2,096,829 5 Nebraska 1,934,408 5 West Virginia 1,792,147 5 Idaho 1,787,065 4 Hawaii 1,415,872 4 New Hampshire 1,359,711 4 Maine 1,344,212 4 Montana 1,068,778 3 Rhode Island 1,059,361 4 Delaware 973,764 3 South Dakota 884,659 3 North Dakota 762,062 3 Alaska 731,545 3 DC 705,749 3 Vermont 623,989 3 Wyoming 578,759 3 Total 328,300,544 538
Can we assume that you are NOT allowing split electoral votes by state (Maine, Nebraska) and you need 270, not 269 electoral votes to win? That's how I'm going about solving the puzzle.
July 27th, 2020 at 4:29:37 PM
permalink
Quote: WizardSame problem but the 2020 U.S. election. Here is a table to help you. Populations as of July 2019. Remember, electoral votes are reset only every ten years.
As a reminder, there is a forum rule against political statements.
State Population Electoral votes California 39,512,223 55 Texas 28,995,881 38 Florida 21,477,737 29 New York 19,453,561 29 Illinois 12,671,821 20 Pennsylvania 12,801,989 20 Ohio 11,689,100 18 Georgia 10,617,423 16 North Carolina 10,488,084 15 Michigan 9,986,857 16 New Jersey 8,882,190 14 Virginia 8,535,519 13 Washington 7,614,893 12 Arizona 7,278,717 11 Massachusetts 6,949,503 11 Tennessee 6,833,174 11 Indiana 6,732,219 11 Missouri 6,137,428 10 Maryland 6,045,680 10 Wisconsin 5,822,434 10 Colorado 5,758,736 9 Minnesota 5,639,632 10 South Carolina 5,148,714 9 Alabama 4,903,185 9 Louisiana 4,648,794 8 Kentucky 4,467,673 8 Oregon 4,217,737 7 Oklahoma 3,956,971 7 Connecticut 3,565,287 7 Utah 3,205,958 6 Iowa 3,155,070 6 Nevada 3,080,156 6 Arkansas 3,017,825 6 Mississippi 2,976,149 6 Kansas 2,913,314 6 New Mexico 2,096,829 5 Nebraska 1,934,408 5 West Virginia 1,792,147 5 Idaho 1,787,065 4 Hawaii 1,415,872 4 New Hampshire 1,359,711 4 Maine 1,344,212 4 Montana 1,068,778 3 Rhode Island 1,059,361 4 Delaware 973,764 3 South Dakota 884,659 3 North Dakota 762,062 3 Alaska 731,545 3 DC 705,749 3 Vermont 623,989 3 Wyoming 578,759 3 Total 328,300,544 538
No votes in California, Texas, Florida, New York, Illinois, Pennsylvania, Ohio, Georgia, North Carolina, Michigan, and Washington. !/2 the votes +1 in each of the other states and DC. So you need around 72,227,750 out of 328,300,544, or around 21.7%
July 27th, 2020 at 4:33:25 PM
permalink
Quote: SOOPOOCan we assume that you are NOT allowing split electoral votes by state (Maine, Nebraska) and you need 270, not 269 electoral votes to win? That's how I'm going about solving the puzzle.
Yes. Please assume all states and DC are winner take all.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 4:44:25 PM
permalink
Quote: SOOPOONo votes in California, Texas, Florida, New York, Illinois, Pennsylvania, Ohio, Georgia, North Carolina, Michigan, and Washington. !/2 the votes +1 in each of the other states and DC. So you need around 72,227,750 out of 328,300,544, or around 21.7%
If you remove Michigan, and Washington and add New Jersey and Virginia, you can win with only 71,587,537 votes and have an electoral vote to spare (in case of an unfaithful elector).
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 27th, 2020 at 7:49:04 PM
permalink
Quote: WizardQuote: gordonm888
The ratio of electoral vote to population increases monotonically from Shire 1 to 10.
So we want the lowest set of shires that add up to 38 elcetoral votes.
Shires 1, 2, 3, 4 add up to 18 electoral votes. We need 20 more. I'll take 6 (8 votes) and 10 (12 electoral votes.)
Shires # 1,2,3,4 6 and 10 have 38/75 electoral votes and 265/560 popular votes: which is 47.3214% of the popular vote.
Gordon, I think you're assuming everyone in any given shire votes the same way. This is not necessarily the case.
Yes I was. An understandable mistake, I live in Tennessee. Given that, I get the same answer as everyone else.
26.877%
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
July 28th, 2020 at 6:58:05 AM
permalink
Quote: Wizard...If you remove Michigan, and Washington and add New Jersey and Virginia, you can win with only 71,587,537 votes and have an electoral vote to spare (in case of an unfaithful elector).
If you rank them by votes needed for each EVote, then that would give 72,038,369 for 273 votes. Then you look at ways to use worse states with fewer votes to get down to 270. There are three obvious ways, but the best are NJ>Wash Miss>Colorado, giving 71,215,374 (21.69%).
