bonanova Posted December 28, 2008 Report Share Posted December 28, 2008 (edited) The eccentric old man decided to plant a new garden. This time, he told his gardener, I want 18 rows of at least 3 trees. But don't waste my money. If you buy more trees than are needed, you'll be fired. But I am a kind man: if the garden includes an additional row - one having 4 trees - I'll give you a raise. [1] How many trees did the careful gardener need to buy? [2] In what configuration did he plant them? [3] Did he get his raise? Edited December 30, 2008 by bonanova remove "at least" from the problem description Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 The eccentric old man decided to plant a new garden. This time, he told his gardener, I want 18 rows of at least 3 trees. But don't waste my money. If you buy more trees than are needed, you'll be fired. But to show my kindness: if you include a 19th row - with 4 trees - I'll give you a raise. How many trees did the careful gardener need to buy, and in what configuration did he plant them? Assuming that trees can be in more than one 'row' at a time (otherwise the answer is trivial). I'd say that 19 is the minimum, configured as shown: __X_X__ X_X_X_X X_X_X_X X_X_X_X X_X_X__ __X_X__ four horizontal rows four vertical rows five diagonal rows top left to bottom right five diagonal rows bottom left to top right Only 18, alas - no magic 19th. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 28, 2008 Author Report Share Posted December 28, 2008 Assuming that trees can be in more than one 'row' at a time (otherwise the answer is trivial). I'd say that 19 is the minimum, configured as shown: __X_X__ X_X_X_X X_X_X_X X_X_X_X X_X_X__ __X_X__ four horizontal rows four vertical rows five diagonal rows top left to bottom right five diagonal rows bottom left to top right Only 18, alas - no magic 19th. The gardener's official statistician has determined that's more than needed. Also, to clarify: a row of 4 trees is only that - it's not 1 or more rows of 3. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 I think it is possible with 15 trees but I don't know how to draw diagrams on Brainden so have represented with letters and numbers in a table for position of trees See table below. The position represented by two dashes does not have a tree. It works if you draw it on graph paper then draw in all the vertical, horizontal and diagonal lines connecting at least three trees – 19 lines in all. The less obvious lines are lines A1-B3-C5 and E1-D3-C5. There are two lines of four trees - diagonal A1-B2-C3-E5 and E1-D2-C3-A5 A1 -- A3 -- A5 -- B2 B3 -- -- C1 C2 C3 C4 C5 -- D2 D3 -- -- E1 -- E3 -- E5 Is this the solution you were looking for? Happy New Year! Alice Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 (edited) The eccentric old man decided to plant a new garden. This time, he told his gardener, I want 18 rows of at least 3 trees. But don't waste my money. If you buy more trees than are needed, you'll be fired. But I am a kind man: if the garden includes an additional row - one having 4 trees - I'll give you a raise. [1] How many trees did the careful gardener need to buy? [2] In what configuration did he plant them? [3] Did he get his raise? A hexagon with three on a side and one in the middle keeps your job at 13 trees. Adding one inline with ONLY one row of three gets you a raise with 14 trees. Edited December 28, 2008 by charles1317 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 I get 18 rows of 3 with 15 trees too, but a different way... row in the middle >4,,, so raise accepted? (dont see how the hexagon works with 13?)GARDNER_PUZZLE.bmp Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 A hexagon with three on a side and one in the middle keeps your job at 13 trees. Adding one inline with ONLY one row of three gets you a raise with 14 trees. How do you get 18 rows of at least 3 trees per row? Alice Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 (edited) How do you get 18 rows of at least 3 trees per row? Alice My solution for a row with four trees is incorrect. It does not provide a nineteenth row. It looks like I don't get a raise. As for the 18 rows with 3 trees each: Edited December 28, 2008 by charles1317 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 28, 2008 Report Share Posted December 28, 2008 My solution for a row with four trees is incorrect. It does not provide a nineteenth row. It looks like I don't get a raise. As for the 18 rows with 3 trees each: Ummm, please disregard. Thats only twelve lines. I wascounting in both directions, not good. Quote Link to comment Share on other sites More sharing options...
0 Prime Posted December 29, 2008 Report Share Posted December 29, 2008 (edited) Here are 19 trees, 18 rows. I don't know if it's the minimum, but it looks pretty. 15 rows parallel to the sides and 3 rows running through the middle of the sides of the hexagon. Edited December 29, 2008 by Prime Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 29, 2008 Report Share Posted December 29, 2008 My original solution above using letters and numbers to represent position of trees - I have now uploaded a photograph. The red spots represent the trees. Quote Link to comment Share on other sites More sharing options...
0 Prime Posted December 29, 2008 Report Share Posted December 29, 2008 Alternatively, If the gardener finds a hill built like a four sided pyramid, he could do 20 rows with only 17 trees, where each of the four hill sides would look like this: Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 29, 2008 Author Report Share Posted December 29, 2008 Amazing creativity has produced the following results: I may have counted wrong here or there; if so let me know. xucam with 19 trees: 5 rows of 2 / 11 rows of 3 / 5 rows of 4 / 2 rows of 6 Alice JH with 15 trees: 15 rows of 3 / 2 rows of 4 / 2 rows of 5 Charles1317 with 13 trees: 12 rows of 3 halfbrain with 15 trees: 19 rows of 3 / 1 row of 5 Prime with 19 trees: 9 rows of 3 / 6 rows of 4 / 3 rows of 5 Prime with 17 trees [and creative landscaping] 20 rows of 3 Halfbrain [with 15 trees] and Prime [with 17 trees] have produced the requisite 18 rows [and more!] of three trees. Both layouts lack the bonus row of [exactly] 4 trees. All layouts so far exceed the minimum number of trees. Here's a hint: Charles missed it by a single tree [and 6 rows.] Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 29, 2008 Report Share Posted December 29, 2008 This is my best shot at it. I'm too tired to count the actual rows right now, but it is 14 trees (I think). At least I can sleep for tonight though . Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 29, 2008 Report Share Posted December 29, 2008 Kern9787 - yes , 14 trees ........but I can only count 17 rows I think?? Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 30, 2008 Author Report Share Posted December 30, 2008 Kern9787: 16 rows of 3, and 1 row of 4, I count. A slight adjustment could create a 17th [vertical] row using 3 of the 4 leftmost points. Still one short.... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 30, 2008 Report Share Posted December 30, 2008 Here's a hint: Charles missed it by a single tree [and 6 rows.] Although suggested minimum number is 12, I post my solution of 13, as I have also a bonus:Black numbers are trees, blue numbers are rows. Row number 15 has 4 trees (bonus) Total 18 rows, 17x3 and 1x4 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 30, 2008 Author Report Share Posted December 30, 2008 Although suggested minimum number is 12, I post my solution of 13, as I have also a bonus:Black numbers are trees, blue numbers are rows. Row number 15 has 4 trees (bonus) Total 18 rows, 17x3 and 1x4 Awesome. This the solution to beat, currently. Nice job. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 30, 2008 Report Share Posted December 30, 2008 (edited) Bonanova - have I misunderstood the question? The wording is ".... I want 18 rows of at least 3 trees." and "...:if the garden includes an additional row - one having 4 trees - I'll give you a raise" so I interpreted this as meaning that a) a total of 19 rows was needed to get the "raise", or "rise" as we call it in the UK b) each of the first 18 rows must contain three or more trees per row (i.e one or two rows could contain more than 3 trees each if necessary) c) the "additional row" i.e. 19th row must contain exactly 4 trees I realised after posting my photo with 15 trees (at top of page 2), that the central tree is not necessary so have posted new photo with 14 trees numbered 1 to 14, and 19 rows (17 x 3 and 2 x 4 ). i.e. there are 18 rows with "at least 3 trees" (17 with exactly 3 trees per row and the 18th with 4). The 19th row contains exactly 4 trees as well! I don't think any of the other answers posted so far, fulfil all the above criteria to get the "raise" or "rise". Heck, these trees are driving me nuts! They must be nut trees, but I don't think either of King Juan Carlos's daughters are likely to visit me! Happy New Year, Alice Edited December 30, 2008 by AliceJH Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 30, 2008 Author Report Share Posted December 30, 2008 Bonanova - have I misunderstood the question? The wording is ".... I want 18 rows of at least 3 trees." and "...:if the garden includes an additional row - one having 4 trees - I'll give you a raise" so I interpreted this as .... Can't believe I did that. Wait. *checks calendar* Yeah, I can believe it. "at least" does not belong there. It doesn't let the bonus row of 4 trees make much sense. Wow - sorry for the headache that caused. Now that several of you have over-achieved the stated OP, shall I show my 18 rows of exactly 3 trees - with a bonus row of 4? Or shall the headaches continue? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 30, 2008 Report Share Posted December 30, 2008 Can't believe I did that. Wait. *checks calendar* Yeah, I can believe it. "at least" does not belong there. It doesn't let the bonus row of 4 trees make much sense. Wow - sorry for the headache that caused. Now that several of you have over-achieved the stated OP, shall I show my 18 rows of exactly 3 trees - with a bonus row of 4? Or shall the headaches continue? Please show @_@ Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 31, 2008 Report Share Posted December 31, 2008 The eccentric old man decided to plant a new garden. This time, he told his gardener, I want 18 rows of at least 3 trees. But don't waste my money. If you buy more trees than are needed, you'll be fired. But I am a kind man: if the garden includes an additional row - one having 4 trees - I'll give you a raise. [1] How many trees did the careful gardener need to buy? [2] In what configuration did he plant them? [3] Did he get his raise? 58 trees unless they were sick trees, in which case, he would've been fired for wasting money anyway 18 rows of 3 and one row of 4 yes, he got his raise Quote Link to comment Share on other sites More sharing options...
0 Guest Posted December 31, 2008 Report Share Posted December 31, 2008 58 trees unless they were sick trees, in which case, he would've been fired for wasting money anyway 18 rows of 3 and one row of 4 yes, he got his raise Welcome to the den dianak! Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted December 31, 2008 Author Report Share Posted December 31, 2008 Please show @_@ Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 3, 2009 Report Share Posted January 3, 2009 Wow. That looks a whole lot simpler than mine. Quote Link to comment Share on other sites More sharing options...
Question
bonanova
The eccentric old man decided to plant a new garden.
This time, he told his gardener, I want 18 rows of at least 3 trees.
But don't waste my money.
If you buy more trees than are needed, you'll be fired.
But I am a kind man: if the garden includes an additional row
- one having 4 trees - I'll give you a raise.
[1] How many trees did the careful gardener need to buy?
[2] In what configuration did he plant them?
[3] Did he get his raise?
Edited by bonanovaremove "at least" from the problem description
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