diff --git a/engine-proto/gram-test/irisawa-hexlet.jl b/engine-proto/gram-test/irisawa-hexlet.jl
new file mode 100644
index 0000000..8b43ad4
--- /dev/null
+++ b/engine-proto/gram-test/irisawa-hexlet.jl
@@ -0,0 +1,77 @@
+include("Engine.jl")
+
+using SparseArrays
+
+# this problem is from a sangaku by Irisawa Shintarō Hiroatsu. the article below
+# includes a nice translation of the problem statement, which was recorded in
+# Uchida Itsumi's book _Kokon sankan_ (_Mathematics, Past and Present_)
+#
+#   "Japan's 'Wasan' Mathematical Tradition", by Abe Haruki
+#   https://www.nippon.com/en/japan-topics/c12801/
+#
+
+# initialize the partial gram matrix
+J = Int64[]
+K = Int64[]
+values = BigFloat[]
+for s in 1:9
+  # each sphere is represented by a spacelike vector
+  push!(J, s)
+  push!(K, s)
+  push!(values, 1)
+  
+  # the circumscribing sphere is internally tangent to all of the other spheres
+  if s > 1
+    append!(J, [1, s])
+    append!(K, [s, 1])
+    append!(values, [1, 1])
+  end
+  
+  if s > 3
+    # each chain sphere is externally tangent to the two nucleus spheres
+    for n in 2:3
+      append!(J, [s, n])
+      append!(K, [n, s])
+      append!(values, [-1, -1])
+    end
+    
+    # each chain sphere is externally tangent to the next sphere in the chain
+    s_next = 4 + mod(s-3, 6)
+    append!(J, [s, s_next])
+    append!(K, [s_next, s])
+    append!(values, [-1, -1])
+  end
+end
+gram = sparse(J, K, values)
+
+# make an initial guess
+guess = hcat(
+  Engine.sphere(BigFloat[0, 0, 0], BigFloat(15)),
+  Engine.sphere(BigFloat[0, 0, -9], BigFloat(5)),
+  Engine.sphere(BigFloat[0, 0, 11], BigFloat(3)),
+  (
+    Engine.sphere(9*BigFloat[cos(k*π/3), sin(k*π/3), 0], BigFloat(2.5))
+    for k in 1:6
+  )...
+)
+frozen = [CartesianIndex(4, k) for k in 1:4]
+
+# complete the gram matrix using Newton's method with backtracking
+L, success, history = Engine.realize_gram(gram, guess, frozen)
+completed_gram = L'*Engine.Q*L
+println("Completed Gram matrix:\n")
+display(completed_gram)
+if success
+  println("\nTarget accuracy achieved!")
+else
+  println("\nFailed to reach target accuracy")
+end
+println("Steps: ", size(history.scaled_loss, 1))
+println("Loss: ", history.scaled_loss[end], "\n")
+if success
+  println("Chain diameters:")
+  println("  ", 1 / L[4,4], " sun (given)")
+  for k in 5:9
+    println("  ", 1 / L[4,k], " sun")
+  end
+end
\ No newline at end of file
diff --git a/engine-proto/gram-test/irisawa-hexlet_bad.jl b/engine-proto/gram-test/irisawa-hexlet_bad.jl
new file mode 100644
index 0000000..8786778
--- /dev/null
+++ b/engine-proto/gram-test/irisawa-hexlet_bad.jl
@@ -0,0 +1,105 @@
+include("Engine.jl")
+
+using SparseArrays
+
+# --- construct the nucleus spheres ---
+
+println("--- Nucleus spheres ---\n")
+
+# initialize the partial gram matrix for the circumscribing and nucleus spheres
+J = Int64[]
+K = Int64[]
+values = BigFloat[]
+for n in 1:3
+  push!(J, n)
+  push!(K, n)
+  push!(values, 1)
+  if n > 1
+    append!(J, [1, n])
+    append!(K, [n, 1])
+    append!(values, [1, 1])
+  end
+end
+gram_nuc = sparse(J, K, values)
+
+# make an initial guess
+guess_nuc = hcat(
+  Engine.sphere(BigFloat[0, 0, 0], BigFloat(15)),
+  Engine.sphere(BigFloat[0, 0, -10], BigFloat(5)),
+  Engine.sphere(BigFloat[0, 0, 11], BigFloat(3)),
+)
+frozen_nuc = [CartesianIndex(4, k) for k in 1:3]
+
+# complete the gram matrix using Newton's method with backtracking
+L_nuc, success_nuc, history_nuc = Engine.realize_gram(gram_nuc, guess_nuc, frozen_nuc)
+completed_gram_nuc = L_nuc'*Engine.Q*L_nuc
+println("Completed Gram matrix:\n")
+display(completed_gram_nuc)
+if success_nuc
+  println("\nTarget accuracy achieved!")
+else
+  println("\nFailed to reach target accuracy")
+end
+println("Steps: ", size(history_nuc.scaled_loss, 1))
+println("Loss: ", history_nuc.scaled_loss[end], "\n")
+
+# --- construct the chain of spheres ---
+
+# initialize the partial gram matrix for the chain of spheres
+J = Int64[]
+K = Int64[]
+values = BigFloat[]
+for a in 4:9
+  push!(J, a)
+  push!(K, a)
+  push!(values, 1)
+  
+  # each chain sphere is internally tangent to the circumscribing sphere
+  append!(J, [a, 1])
+  append!(K, [1, a])
+  append!(values, [1, 1])
+  
+  # each chain sphere is externally tangent to the nucleus spheres
+  for n in 2:3
+    append!(J, [a, n])
+    append!(K, [n, a])
+    append!(values, [-1, -1])
+  end
+  
+  # each chain sphere is externally tangent to the next sphere in the chain
+  #=
+  a_next = 4 + mod(a-3, 6)
+  append!(J, [a, a_next])
+  append!(K, [a_next, a])
+  append!(values, [-1, -1])
+  =#
+end
+gram_chain = sparse(J, K, values)
+
+if success_nuc
+  println("--- Chain spheres ---\n")
+  
+  # make an initial guess, with the circumscribing and nucleus spheres included
+  # as frozen elements
+  guess_chain = hcat(
+    L_nuc,
+    (
+      Engine.sphere(10*BigFloat[cos(k*π/3), sin(k*π/3), 0], BigFloat(2.5))
+      for k in 1:6
+    )...
+  )
+  frozen_chain = [CartesianIndex(j, k) for k in 1:3 for j in 1:5]
+  
+  # complete the gram matrix using Newton's method with backtracking
+  L_chain, success_chain, history_chain = Engine.realize_gram(gram_chain, guess_chain, frozen_chain)
+  completed_gram_chain = L_chain'*Engine.Q*L_chain
+  println("Completed Gram matrix:\n")
+  display(completed_gram_chain)
+  if success_chain
+    println("\nTarget accuracy achieved!")
+  else
+    println("\nFailed to reach target accuracy")
+  end
+  println("Steps: ", size(history_chain.scaled_loss, 1))
+  println("Loss: ", history_chain.scaled_loss[end], "\n")
+end
\ No newline at end of file