Trees in lines

Problems this week are rated on 1 5 star scale.

  1. The evil castle lord and the young gardener
  2. **/*****
    Let’s start with a small problem in this category of my own design:
    An evil castle lord wanted to fire the handsome young gardener who was eyeing his only daughter. “There are already two trees in the orchard-to-be,” he told the young lad, and here are five more. That makes a total of seven trees. Plant them in 6 rows of three trees each.”
    The evil castle lord handed him a sketch how 7 trees could be planted in 6 rows of 3 each:
    7 trees in 6 rows of 3 exercise A

    “There is a watchtower,” the castle lord warned him, “which none of the lines of 3 may cross. Also, neither of the two trees may not be moved.”

    The young gardener went to the future orchard, and to his chagrin found that the sketch of the castle lord was complete useless, as a heavy watchtower had been erected between the two trees, which were along one of the sides of the orchard:
    7 trees in 6 rows of 3 exercise B

    Luckily, the kind daughter of the castle lord came to his rescue and suggested a layout which fulfilled her fathers requirements, left the watchtower untouched and which didn’t need the two existing trees to be moved.

    You can check your solutions here

    Note that problems of his type have been featured before:

  3. *
  4. 9 trees***/*****
    For the really tough ones, we will have to turn to the 19th century masters, aka Henry Dudeney. Henry calls them “point and line problems”. According to Dudeney, the oldest problem goes back to the great Sir Isaac Newton:
    Plant 9 trees in such a way that there are 10 straight rows with 3 trees each.

    You can check your solutions here

  5. After Sir Isaac Newton, John Jackson is the next person to pose problems in this category. In his book Rational Amusement for Winter Evenings, published in 1821, John Jackson gives 10 examples of these problems. I will copy the problems here, though I do not have the solutions:

  6. John Jackson no 1***/*****
    Your add I want. nine trees to plant
    in rows just half a score;
    and let there be in each row three
    solve this: I ask no more.
  7. John Jackson no 2***/*****
    Fain would I plant a grove in rows,
    but how must I its form compose
    with three trees in each row
    to have as many rows as trees
    now tell me, artists, if you please,
    ‘t is all I want to know.
  8. John Jackson no 3****/*****
    Ingenious artists, if you please
    to plant a grove, pray show,
    in twenty three rows with fifteen trees,
    and three in every row
  9. John Jackson no 4****/*****
    It is required to plant 17 trees in 24 rows
    and to have 3 trees in every row
  10. John Jackson no 5****/*****
    Ingenious artists, pray dispose
    twenty four trees in twenty four rows
    three trees I’d have in every row
    a pond in the midst I’s have also
    a plan thereof I fain would have
    and therefor you assistance crave
  11. John Jackson no 6****/*****
    Fam’d arborists, display your power,
    and show how I may plant a bower
    with verdant fir and yew:
    twelve trees of each I would dispose
    in only eight-and-twenty rows;
    four trees in eacht to view.
  12. John Jackson no 7****/*****
    Plant 27 trees in 15 rows, 5 in a row.
  13. John Jackson no 8*****/*****
    Ingenious artists, if you please,
    now plant me five-and-twenty trees,
    in twenty-eight rows, nor less, nor more;
    In some rows five, some three, some four.
  14. John Jackson no 9*****/*****
    It is required to plant 90 trees in 10 rows,
    with 10 trees in each row; each tree equidistant
    from the other, also each row equidistant from a pond in the center
  15. John Jackson no 10*****/*****
    A gentleman has a quadrangular irregular piece of ground, in which he is desirous of planting a quincunx, in such a manner, that all the rows of trees, whether transversal or diagonal shall be right lines. How must this be done?

    Note: A real quincunx is a plantation of trees disposed in a square. consisting of 5 trees, one at each corner, and the fifth in the middle; but in the present case, the trees are to be disposed in a quadrangle, one at each corner (as in the square) and the fifth at the intersection of the two diagonals.

  16. Though the book with the problems is readily available on the internet, the solutions have been drawn in a couple of plates that alas are not included with the scans of the book.
    Here’s a comprehensive table of John’s problems:

    Nr coins/trees lines rowlength solution
    1 9 10 3 solution
    2 x x 3 solution
    3 15 23 3
    4 17 24 3
    5 24 24 3
    6 2*12 28 4
    7 27 15 5
    8 25 28 3, 4 or 5
    9 90 10 equidistant rows

    Dudeney gives 5 problems in his Amusement in Mathematics

  17. The king and the castles****/*****
    There was once, in ancient times, a powerful king, who had eccentric ideas on the subject of military architecture. He held that there was great strength and economy in symmetrical forms, and always cited the example of the bees, who construct their combs in perfect hexagonal cells, to prove that he had nature to support him. He resolved to build ten new castles in his country all to be connected by fortified walls, which should form five lines with four castles in every line. The royal architect presented his preliminary plan in the form I have shown. But the monarch pointed out that every castle could be approached from the outside, and commanded that the plan should be so modified that as many castles as possible should be free from attack from the outside, and could only be reached by crossing the fortified walls. The architect replied that he thought it impossible so to arrange them that even one castle, which the king proposed to use as a royal residence, could be so protected, but his majesty soon enlightened him by pointing out how it might be done. How would you have built the ten castles and fortifications so as best to fulfill the king’s requirements? Remember that they must form five straight lines with four castles in every line.
    dudeney king and castles q206

  18. You can check your solutions here

  19. Cherries and plums**/*****
    dudeney q207

  20. The illustration is a plan of a cottage as it stands surrounded by an orchard of fifty-five trees. Ten of these trees are cherries, ten are plums, and the remainder apples. The cherries are so planted as to form five straight lines, with four cherry trees in every line. The plum trees Pg 57are also planted so as to form five straight lines with four plum trees in every line. The puzzle is to show which are the ten cherry trees and which are the ten plums. In order that the cherries and plums should have the most favourable aspect, as few as possible (under the conditions) are planted on the north and east sides of the orchard. Of course in picking out a group of ten trees (cherry or plum, as the case may be) you ignore all intervening trees. That is to say, four trees may be in a straight line irrespective of other trees (or the house) being in between. After the last puzzle this will be quite easy.

    You can check your solutions here

  21. A plantation puzzle**/*****
    dudeney q208

  22. A man had a square plantation of forty-nine trees, but, as will be seen by the omissions in the illustration, four trees were blown down and removed. He now wants to cut down all the remainder except ten trees, which are to be so left that they shall form five straight rows with four trees in every row. Which are the ten trees that he must leave?

    You can check your solutions here

  23. The twenty-one trees***/*****
    A gentleman wished to plant twenty-one trees in his park so that they should form twelve straight rows with five trees in every row. Could you have supplied him with a pretty symmetrical arrangement that would satisfy these conditions?

    You can check your solutions here

  24. The ten coins***/*****
    dudeney q210 ten coins

  25. Place ten pennies on a large sheet of paper or cardboard, as shown in the diagram, five on each edge. Now remove four of the coins, without disturbing the others, and replace them on the paper so that the ten shall form five straight lines with four coins in every line. This in itself is not difficult, but you should try to discover in how many different ways the puzzle may be solved, assuming that in every case the two rows at starting are exactly the same.

    You can check your solutions here

  26. The twelve mince-pies**/*****
    dudeney q211 the twelve mince-pies

  27. It will be seen in our illustration how twelve mince-pies may be placed on the table so as to form six straight rows with four pies in every row. The puzzle is to remove only four of them to new positions so that there shall be seven straight rows with four in every row. Which four would you remove, and where would you replace them?

    You can check your solutions here

  28. The Burmese plantation**/*****
    A short time ago I received an interesting communication from the British chaplain at Meiktila, Upper Burma, in which my correspondent informed me that he had found some amusement on board ship on his way out in trying to solve this little poser.

    dudeney q212 burmese plantation

  29. If he has a plantation of forty-nine trees, planted in the form of a square as shown in the accompanying illustration, he wishes to know how he may cut down twenty-seven of the trees so that the twenty-two left standing shall form as many rows as possible with four trees in every row.

    Of course there may not be more than four trees in any row.

    You can check your solutions here

  30. Turks and Russians**/*****
    This puzzle is on the lines of the Afridi problem published by me in Tit-Bits some years ago.

    On an open level tract of country a party of Russian infantry, no two of whom were stationed at the same spot, were suddenly surprised by thirty-two Turks, who opened fire on the Russians from all directions. Each of the Turks simultaneously fired a bullet, and each bullet passed immediately over the heads of three Russian soldiers. As each of these bullets when fired killed a different man, the puzzle is to discover what is the smallest possible number of soldiers of which the Russian party could have consisted and what were the casualties on each side.

    You can check your solutions here

Notes and sources

  1. Elkies, Noam D research some problems in this field and published them in the Periodica Mathematica Hungarica 53, (1-2): 133-148.

  2. Planting trees, by Stefan Burr, city university of New York

  3., Arrangements of lines, by Brian Harbour and Tomasz Szemberg

  4. Orchard-Planting Problem

  5. On some points-and-lines problems and configurations, by Noam D. Elkies

  6. Martin Gardner in scientific american: The symmetrical arrangement of the stars on the American flag and related matters
  7. Puzzles from other worlds, by Martin Gardner

Finally, I’d like to thank Stephen Flaherty for helping me with finding some of the papers listed above on this subject. The available time prevented me to use all the research in this in this post. I hope to use this in a subsequent post.


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