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This puzzle is the new series after

You are given the following map of Africa:

Map of All African Countries

And the following list of 54 African countries:

  1. Algeria
  2. Angola
  3. Benin
  4. Botswana
  5. Burkina Faso
  6. Burundi
  7. Cabo Verde
  8. Cameroon
  9. Central African Republic
  10. Chad
  11. Comoros
  12. Democratic Republic of the Congo
  13. Republic of the Congo
  14. Djibouti
  15. Egypt
  16. Equatorial Guinea
  17. Eritrea
  18. Eswatini
  19. Ethiopia
  20. Gabon
  21. Gambia
  22. Ghana
  23. Guinea
  24. Guinea-Bissau
  25. Ivory Coast
  26. Kenya
  27. Lesotho
  28. Liberia
  29. Libya
  30. Madagascar
  31. Malawi
  32. Mali
  33. Mauritania
  34. Mauritius
  35. Morocco
  36. Mozambique
  37. Namibia
  38. Niger
  39. Nigeria
  40. Rwanda
  41. Sao Tome and Principe
  42. Senegal
  43. Seychelles
  44. Sierra Leone
  45. Somalia
  46. South Africa
  47. South Sudan
  48. Sudan
  49. Tanzania
  50. Togo
  51. Tunisia
  52. Uganda
  53. Zambia
  54. Zimbabwe

Using the above map, you must place triangles such that all 54 countries have exactly one triangle on it. It is fine if the same triangle leaves and reenters the same country.

How many triangles do you need at minimum? A trivial non-optimized answer is 54 triangles, one per country! However, you can use one triangle to touch Morocco, Tunisia, Algeria and more.

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  • $\begingroup$ There is a border shown between Morocco and Western Sahara, which is not listed. $\endgroup$ Commented Mar 5 at 7:59
  • $\begingroup$ @WeatherVane Western Sahara is in a weird undefined state in which neither its independence nor Morocco's claim is widely recognized, while Spain no longer claims it. On maps it's often represented as not belonging to any country, or as administered by Morocco but separated by a border. Including the border but not listing it as a country is fairly standard. If the OP hasn't listed it, it probably doesn't have to be included, but the optimum will almost certainly touch it anyway. $\endgroup$ Commented Mar 5 at 13:39

2 Answers 2

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A solution with 4 triangles

Map of Africa

The circles around the 5 islands are not supposed to be an area – only to highlight them. In fact, of all the islands in each group, I used only the largest.

Western Sahara, and Ceuta / Melilla (two Spanish enclaves) are ignored.

The problem always is deciding exactly where the borders are; they are just smeared lines. Worse, several countries have disputed borders, some with two marked on the OP's map. I settled the disputes unilaterally (at random) for the purposes of the puzzle, before solving the triangles.

The map is derived from the OP's map. I identified the countries algorithmically, and replaced any border with the majority of adjacent pixels, then applied a four-colour algorithm.

The lines were placed by hand, using a tool I made to drive the vertices around, check the coverage, and identify multiple crossings of any country (with Bresenham pixel stepping). One vertex is off the map in the Indian Ocean.

With the original map:

Original Africa map marked up

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  • $\begingroup$ I think that this is correct, but how can you assert that the triangle over Madagascar isn't touching DRC and is touching both Zimbabwe and Malawi instead of narrowingly missing? I tried to draw that in Google Maps and it works, but Google Maps considers the Earth curvature, differently than this map. $\endgroup$ Commented Mar 10 at 16:21
  • $\begingroup$ @VictorStafusa obviously the projection will make it different. If you download my maps so you can zoom in, you should be able to to see clear land each side of a line somewhere in a state, and for near-misses you can see that the line does not even graze a border, again there is clear land colour to be seen in between. $\endgroup$ Commented Mar 10 at 16:32
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    $\begingroup$ Oh, I see. Well done, bravo! $\endgroup$ Commented Mar 10 at 16:49
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Got a solution with 5 triangles:

5 triangles

The yellow triangle extends long into the Atlantic Ocean.

Also:

Oh, Western Sahara!

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  • $\begingroup$ I did a lot of trying to get a 4-triangle solution and I gave up. I think that no 4-triangle solution exists. Uganda, Rwanda, Burundi, Lesotho, Eswatini, Djibouti, Equatorial Guinea, Gambia, Guinea, Tunisia and the island countries are targets that must be either very precisely hit or very thinly avoided and that restricts the possible solutions and frustrate many ideas of solving this using less than 5 triangles. $\endgroup$ Commented Mar 7 at 4:27

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